A Fuzzy Cognitive Map for M

exico City’s Water Availability System

an Paz Ortiz and Carlos Gay Garc

ıa

Programa de Investigaci

on en Cambio Clim

atico, Universidad Nacional Aut

onoma de M

exico, Mexico City, Mexico

Keywords:

Cognitive Maps, Water System, Climatic Scenarios, Fuzzy Analysis.

Abstract:

In the present work a fuzzy cognitive map was used to analyze M

exico city’s water system, to study the water

availability and the system’s response under different possible scenarios of climate change. The map includes

the water sources and their availability, as well as climatic and social factors that affect the system. The map

was built based on the analysis of the previous study “Vulnerabilidad de las fuentes de abastecimiento de

agua potable de la Ciudad de M

exico en el contexto de cambio clim

atico.” (“Vulnerability of fresh water

sources in Mexico city in the context of climate change”) by Escolero (2009). Once the map was built, it was

analyzed using the technique of vector state and adjacent matrix. First, the values of {0,1} were used to ﬁnd

the “hidden patterns”. Then, different weights were considered for the edges to analyze the system’s sensibility

to changes in the strength of the processes. Finally, to investigate the importance of different nodes over the

water availability, the min - max criteria was used to propose implementations for possible solutions. In the

analysis, considerations were made between climatic and social drivers, in order to assign the corresponding

attributions for each kind.

1 INTRODUCTION

1.1 Fuzzy Cognitive Maps (FCMs)

FCMs use graph structures to represent the ﬂux

of cause and effect relationships among prede-

ﬁned variable concepts; these are depicted as nodes

, C

, ..., C

) in an interconnected network, each

one representing a concept, and the edges e

i j

which

connect two nodes (denoted as C

→ C

) are causal

connections and they express how much C

causes C

These edges can be negative or positive. A positive

relation C

→

states that if C

grows also does

, and a negative relation C

¯→ C

indicates that as C

grows, C

decreases. The edges among the network

nodes can be represented in a an adjacent matrix, and

the state of each node at an speciﬁc time can be rep-

resented in a row vector state. The value of the vector

for each iteration is calculated by multiplying the vec-

tor by the adjacent matrix.

= v

t−1

∗ M (1)

FCMs can be used to model the causal relation-

ships of a system. To accurately capture the system’s

dynamic, the maps are usually built considering ex-

perts opinions (Kosko 1986). Once they are built, they

can be analyzed with different techniques that give in-

formation about the system’s properties. We consider

a map fuzzy when we have used linguistic quantiﬁers

as weights for its edges (Kosko 1992).

1.2 Fuzzy Sets

The fuzzy sets are classes of objects, taken from a uni-

verse, with a continuum of grades of membership to

a particular set (Zadeh, 1965). These grades of mem-

bership are speciﬁed by a membership function which

assigns a degree of membership to each fuzzy subset

for each object. The fuzzy sets are a useful tool to

work in universes where we have imprecision in the

class membership criteria, i.e. when it is not well de-

ﬁned whether or not an element belongs to an spe-

ciﬁc set. In our study, we will use fuzzy sets to model

causality when it is referred by experts as linguistic

quantiﬁers, such as low or high.

1.3 Indirect and Total Effect (min - max

criteria)

As the cognitive maps represent the causal ﬂux among

the nodes of the network, one thing that is important

to know is how much causality a node imparts over

another. To know this, we use the indirect and total

495

Paz Ortiz I. and Gay García C..

A Fuzzy Cognitive Map for México City’s Water Availability System.

DOI: 10.5220/0004620704950503

In Proceedings of the 3rd International Conference on Simulation and Modeling Methodologies, Technologies and Applications (MSCCEC-2013), pages

495-503

ISBN: 978-989-8565-69-3

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

causal effects when we work with linguistic edges.

A causal path from some concept C

to concept C

comprises the sequence C

→C

. . . →C

→C

. The

indirect effect of C

on C

is the causality C

imparts

to C

via the path (i, k

. . . k

, j). The total effect of

on C

is the composite of all the possible indirect

effects that C

imparts over C

. The indirect effect of

an speciﬁc branch or path is deﬁned as the minimum

value of all the edges in the path. And the total effect

is deﬁned as the maximum of all the indirect effects

(Kosko, 1986).

As the causality can be positive or negative, we

used the following rules in order to know the sign of

the causality of one node over another. The indirect

effect of a path is negative if the number of negative

edges in the path is odd, and is positive if the number

is even. The total effect of C

on C

is negative if all

the indirect effects of C

on C

are negatives, and is

positive if they are all positive, in any other case the

total effect is undetermined. The indeterminacy can

be removed using a weighted scheme. If we assign

weights to the edges (this weights can be positive or

negative real numbers) w

i j

then the indirect effect in

a path (i, k

. . . k

, j) is the product w

i,k

∗ w

i,k

∗ . . . ∗

i,k

and the total effect is the sum of all path products

(Pel

aez 1996).

2 THE MODEL

2.1 State of the Art

Soft computing models oriented to systems analysis

and decision making have gained popularity in differ-

ent areas. The utility of these kind of models in com-

parison with hard computing models relies on their

tolerance for imprecision and their ability to make de-

cisions under uncertainty (Nguyen et al. 2003). Fuzzy

cognitive maps were introduced by Kosko (1986) and

since then, they have been used to model complex

systems (Stylios 2004), distributed systems (Stylos

et al. 1997), as a system model for failure modes

and effects analysis (Pel

aez 1996), and to model vir-

tual worlds (Kosko 1994). In environmental sciences,

they have been recently used for environmental deci-

sion making and management (Elpiniki 2012), and to

evaluate cases of study, like the future of water in the

Seyhan Basin (Cakmak 2013) and the description of

current system dynamics together with the develop-

ment of land cover scenarios in the Brazilian Amazon

(Soler et al. 2010).

The water availability problem is one of the many

ﬁelds where we need to work under great uncertainty.

The climate models have different outputs. Models

should deal with social, climatic and political vari-

ables, and many of these processes are not precisely

quantiﬁed. Then, the FCMs are a useful tool to ex-

plore, assess and make strategic decisions.

2.2 The Fuzzy Cognitive Map

The FCM was built throughout a detailed analysis of

the study “Vulnerabilidad de las fuentes de abastec-

imiento de agua potable de la Ciudad de M

exico en

el contexto de cambio clim

atico.” (“Vulnerability

of fresh water sources in Mexico city in the context

of climate change”) by Escolero (2009). First, the

concepts that describe the dynamics of the system

were identiﬁed, and these were related in terms

of causes and consequences. When the map was

ﬁnished, Dr. Escolero was consulted to validate the

map, and to add relations and nodes that were missed.

The concepts identiﬁed in the study were divided

in climatic and social variables. This classiﬁcation

allowed us to have a more detailed analysis of the

system. The following concepts were identiﬁed:

• Climatic concepts: Temperature increase, which

refers to the increase in mean temperature and in

the extreme values; Precipitation (PCP) decrease,

which refers to the decrease in mean precipitation;

and Extreme events increase, that refers to the rise of

the frequency and intensity of extreme events (IPCC

2007). These three concepts belong to climatic

variables, and so, they represented the climate drivers

on the map.

• Social identiﬁed concepts are: Population growth,

describes the increase of inhabitants in the city;

Agriculture development, due to land use change

and increases the water demand; Urbanization, refers

to the development and growth of the city; and

Industry growth, that refers to the development of the

industries and its possible development tendencies.

As consequences of the drivers mentioned above,

the result concepts are: Waste water discharge

increase, Degradation in the water quality, Vegetal

cover loss, Social water demand increase, Erosion

increase, Non-planned extraction, Evapotranspiration

increase, Aquifers degradation referring to basins,

Basins degradation in the case of dams, Decrease in

dams levels, Decrease in water availability (which is

the concept we wanted to analyze), and ﬁnally, Social

and administrative conﬂicts increase. The fuzzy

cognitive map for M

exico’s city water availability

system is shown in Figure 1.

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Figure 1: Fuzzy Cognitive map for M

exico city’s water

availability system.

3 ANALYSIS

3.1 Analysis of Feedback Processes,

Social and Climate Drivers

Our ﬁrst analysis identiﬁed the feedback processes in

the map. The feedbacks are important because they

are the parts of the network that once they are ini-

tiated, the nodes constituting them will continue in-

teracting, even though the drivers are turned off. We

identiﬁed four feedbacks in the map:

• The ﬁrst one is between the nodes 12 and 14, and it

represents the process between erosion increase and

the basins degradation (in the Cutzamala). A simi-

lar structure is between nodes Aquifers degradation

(Basins, 15) and Aquifer recharge decrease (17).

• The third process is constituted by nodes 13 (non

planned extraction), 15 (aquifers degradation basins),

17 (aquifer recharge decrease), 18 (decrease in water

availability), and 19 ( social and administrative con-

ﬂicts increase).

• The fourth one is among concepts 13, 16, 18, 19,

and 13.

When any node in a feedback is activated, the pres-

ence of the signal remains in the map, and only dis-

appear if it is damped by fractionary weights at the

edges.

We analyzed the map using the technique of the

adjacent matrix and the vector state described section

1. We iterate using equation 1. We will denote de t-th

iteration by v

In order to check the feedback processes we

started climate and social drivers separately. In the

ﬁrst case, starting climate drivers i.e. Temperature

increase and PCP decrease (the initial vector had a 1

in its ﬁrst and second entrances), without considering

in this case extreme events, the system reached the

ﬁnal vector in ﬁve iterations:

= [ 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 0 ]

Which means that together the increase of tem-

perature and the PCP decrease, will trigger nodes

13 (non-planned extraction), 15 (aquifers degrada-

tion, basins), 16 (decrease in dams levels), 17 (aquifer

recharge decrease), 18 (decrease of water availabil-

ity), and 19 (social and administrative conﬂicts in-

crease).

When nodes 3 (population growth), 4 (agriculture

development), 5 (urbanization), and 6 (industry

growth) were activated, the system converged in ﬁve

iterations to the vector:

= [ 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 ]

Which differs from our last result only by the

nodes 12 (erosion increase) and 14 (basins degrada-

tion, Cutzamala). When we considered C

→ C

the

resulting vectors were equals.

In both cases, we turned on the forcers only in the

initial vector (v

) and we let the system evolve. As

expected, the nodes with feedback remained ON even

though the driver was turned OFF.

We observed that the resulting vectors, forcing cli-

mate or social nodes, differ only by two nodes. This

indicates that nodes 13 (non-planned extraction), 15

(aquifers degradation basins), 16 (decrease in dams

levels), 17 (aquifer recharge decrease), 18 (decrease

of water availability), and 19 (social and administra-

tive conﬂicts increase) are driven by climate and so-

cial factors.

3.2 Hidden Patterns

Once we identiﬁed the feedback processes, we ana-

lyzed the map in the cases where we kept the forcing

after each iteration. These cases showed the ﬁnal

states, when the forcer remained in the system. For

this analysis we considered the state vector and the

AFuzzyCognitiveMapforMéxicoCity'sWaterAvailabilitySystem

497

adjacent matrix with values in the set {0, 1}. Forcing

the system to different conﬁgurations showed how

the system responds to various initial conditions. We

used the three following conﬁgurations:

a. First we turned ON the Temperature increase and

PCP decrease and turned OFF the concepts 3, 4,

5 and 6. This showed how the system responds to

persistent climate forcing.

b. Turned OFF the Temperature increase and the

PCP decrease, and turned ON the concepts 3, 4, 5

and 6. Which considers only the social forcing of the

system.

c. We turned the Temperature increase and PCP

decrease ON, as well as concepts 3, 4, 5, and 6.

Which showed the system’s behavior having both

climate and social forcing.

In the ﬁrst case (a), turning ON the nodes 1 and 2

(Temperature increase and PCP decrease), the system

converged in three iterations to the vector:

= [ 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 1 1 ]

This last result means that keeping nodes 1 and 2

forced increases the concepts: water social demand,

non-planned extraction, aquifers degradation (in

basins), decrease in dams levels, aquifer recharge

decrease, decrease in water availability, and social

and economic conﬂicts.

When node 11 was turned ON (extreme events

increase) together with nodes 1 and 2, the system

converged in three iterations to:

= [ 1 1 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1 ]

With this conﬁguration, nodes 12 (erosion in-

crease), and 14 (basin degradation, Cutzamala) were

turned ON.

In the case (b), when we turned OFF the nodes 1,

2, and 11, but instead we turned ON the nodes 3, 4, 5,

and 6, the system converged in four iterations to:

= [ 0 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 ]

Nodes 7 (waste waters discharge increase), 8

(vegetal cover loss), were turned ON and node 10

(degradation in water quality), and node 20 (evapo-

transpiration increase) were turned OFF.

Case (c), we turned ON the concepts 1, 2, 3, 4, 5,

6, and 11. The system converged in three iterations to

the vector:

= [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ]

The difference between only considering temper-

ature increase (1) and PCP decrease (2) in compari-

son of turning ON the nodes population growth (3),

agriculture development (4), urbanization (5), and in-

dustry growth (6), both keeping the forcer, are basi-

cally the nodes: waste waters discharge increase (7),

vegetal cover loss (8), degradation in the water qual-

ity (10), erosion increase (12) and basins degradation,

Cutzamala (14). Notice that this nodes were turned

on when social forcers were ON.

3.3 Considering Climate Predictions of

Models

Although the climate drivers on M

exico city do not

completely depend on the processes developed there

(over the city), we know how they change and in-

teract by following the predictions given by the cli-

mate models (Oglesby et al. 2010). Different pre-

dictions allowed us to use the model we created to

explore different scenarios. The climate forcers in the

model are: temperature increase, PCP decrease, and

extreme events increase. The relations among these

three concepts, considering the scenarios of precipita-

tion decrease, and its opposite, precipitation increase,

are shown in the next image (Figure 2).

Figure 2: Interaction among climate forcers considering the

scenarios of precipitation decrease and increase.

As we said, the scenarios stated by the models

suggest different relations between the nodes tem-

perature increase and PCP decrease. Some models

suggest that if the temperature increases, the amount

of precipitation will decrease, while other models

suggest the opposite (Oglesby et al 2010). So, in the

map we considered positive and negative causality

separately for different scenarios. On the other hand,

all the models predicted an increase in extreme

events, but they ranged in their predictions among

intensity and frequency of the events. So the edge

from temperature increase to extreme events increase

is considered to be positive. If we assume that an

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increase in temperature causes an increase in the

PCP decrease and an increase in extreme events,

and running the model with these characteristics, i.e.

connecting node 1 with nodes 2 and 11, and restarting

node 1 after each iteration we have in four iterations

the following vector:

∗

= [ 1 1 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1]

Which in comparison with the vector v

obtained

in the case (a) in section 3.2, when nodes 1 and 2

were forced, differs in nodes 11, 12 and 14.

= [ 1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 1 1 1 ]

Considering that an increase in temperature

causes a decrease in PCP decrease (i.e. an increase in

PCP), in ﬁve iterations we have:

= [1 -1 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 1 1]

As seen, the only difference with respect to v

∗

the −1 in the node 2. Which means that even though

we considered an increase in PCP, the structured

interactions in the map have kept the tendency. This

is due to the fact that many nodes have more than one

node which is activating them, therefore, the sum of

the interactions is more than one, before the threshold

function.

With our approach of 0 and 1 for edges and values,

this is as far as we can go. But is very intuitive that

different processes have different amounts of causal-

ity, e.g. we know that temperature increase causes

social water demand increase, as well as evapotran-

spiration increase, but the strength of causality could

be different. It is convenient at this point to assign

pondered weights for each process in the map.

It is important to emphasize that we are evaluating

the map in a qualitative way, and assigning weighted

values only indicates a degree of causality among

concepts. We are not assuring, by indicating 0.5 for

a speciﬁc edge, that one concept causes exactly 0.5

the other. But, pondering the edges we gain intuition

about the map’s importance and sensitivity to each

process. These weighted values can be assigned by

expert’s opinion or proposed assuming speciﬁc con-

ditions we want to analyze.

3.4 Considering weighted Edges for

Sensitivity Analysis

In the previous analysis, we only considered positive

or negative causality in order to ﬁgure out the gen-

eral behavior of the system. However, this assump-

tion requires each edge to be either positive or neg-

ative, but with the same causality value. Neverthe-

less, we know that different processes have different

amounts of causality, for example, it could be intu-

itive that the temperature increase will have different

impacts in social water demand increase and in evapo-

ration increase, and this is an interesting thing to ana-

lyze. As the next example will show, these changes in

the value of interactions will lead different ﬁnal states

for different edges values. Then, when we consider

weighted edges, the map’s behavior is qualitatively

different. In this case, some processes occur slowly

and some times their values will stabilize in a differ-

ent point.

For example, the following vector shows the state vec-

tor resulting from considering a value of 0.5 for each

edge after 10 iterations, when the system converges

into an equilibrium state given by:

= [ 1 0.5 0 0 0 0 0 0 0.5 0 0.5 0.33 0.44 0.16

0.61 0.80 0.80 0.79 0.39 0.5]

We can see how different nodes reached different

values in a new equilibrium state of the system. As

we said, we have considered all edges with value of

0.5 and we have restarted v[1] = 1 after each iteration.

In the Figure 3, the upper graph shows the evolution

of the concepts: erosion increase(12), non-planned

extraction (13), basins degradation (Cutzamala) (14),

aquifers degradation (basins) (15), decrease in dams

levels (16), aquifer recharge decrease (17), decrease

water availability (18), and social and administrative

conﬂicts increase (19). Concepts 3, 4, 5, 6, and 7

(which correspond with social forcers) were turned

off. Concepts 8 and 10 were not activated, because

they depend on the social forcers. Finally, the value

of C

(1), remained equal to 1, and the concepts 2, 9,

11 and 20 were not shown, because as they are only

activated by the concept C

, they remained with value

of 0.5. When upper and bottom graphs are compared,

it can be observed how dependent is the ﬁnal value

of the concepts with the value of the edges. In the left

graph we used the same conﬁguration but we changed

the sign in the edge C

→ C

which is taken as −0.5.

In this case the state vector after 10 iterations is:

= [ 1 -0.5 0 0 0 0 0 0 0.5 0 0.5 0.33 0.28 0.16

0.18 0.22 0.09 0.15 0.07 0.5]

These results show that the model is highly sen-

sible to changes in the edge values, sings, and initial

conditions, however we can see that the concept’s val-

ues tendency to growth remains.

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499

Figure 3: Upper graph: evolution of the concepts erosion

increase(12), non-planned extraction (13), basins degrada-

tion (Cutzamala) (14), aquifers degradation (basins) (15),

decrease in dams levels (16), aquifer recharge decrease (17),

decrease water availability (18), and social and administra-

tive conﬂicts increase (19) after 10 iterations with all the

edges having a value of 0.5. Bottom: the same conﬁgura-

tion, but changing edge C

→ C

which is taken as −0.5. In

both graphs concept C

is in pink color.

3.5 Using weighted Edges

As we just discussed, the behavior of the model is

completely different depending on the value of the

edges. Nevertheless, sometimes is difﬁcult to ponder

processes of different nature, like the increase in so-

cial and administrative conﬂicts and the temperature

increase. For this reason, this process involves the

experts opinion. We assigned the value of the edges

by consulting Dr. Escolero. We asked him to assign

“how much one node causes another” using linguistic

quantities. In this case, we chose as linguistic quan-

tities: low (L), medium (M), and high (H). These lin-

guistic values are shown in Figure 1.

This map contains the expert opinion, which is

based on the knowledge of the system. Now, in order

to analyze it using the technique of the adjacent ma-

trix and the vector sate, we divided the interval [0,1]

in three subsets, each one of them representing one

set of strength of causality. Then, we chose a rep-

resentative value for each set which was used as its

corresponding entrance in the matrix. In this case we

chose the values of 0.3 for low, 0.6 for medium, and

0.9 for high. And we selected medium causality for

the edge C

→ C

and low for C

→ C

Then, we ran the model using these values,

starting only C

with value of 1 and keeping the

forcing. The state vector after six iterations is:

= [1 0.9 0 0 0 0 0 0 0.6 0 0.3 0.16 1 0.08 1 1 1

1 0.9 0.9]

Concepts 2 (0.9 PCP decrease), 9 (0.6 social in-

crease in water demand), 11 (0.3 extreme event in-

crease), and 20 (0.9 evapotranspiration increase) re-

mained constant, as they were activated by C

. Con-

cepts 3, 4, 5, and 6 (social forcers) were not acti-

vated. Concepts 7, 8, and 10 depended on the so-

cial forcers. In ﬁgure 6, we represented concepts ero-

sion increase(12), non-planned extraction (13), basins

degradation (Cutzamala) (14), aquifers degradation

(Basins) (15), decrease in dams levels (16), aquifer

recharge decrease (17), decrease water availability

(18), and social and administrative conﬂicts increase

(19) after 6 iterations.

In Figure 4 Upper, concepts 12 (erosion increase)

and 14 (basins degradation, Cutzamala), which form a

feedback cycle, stabilize at 0.16 and 0.08 respectively,

as well as concept 19 social and administrative con-

ﬂicts increase with a value of 0.9. All other concepts

exceeded the value of one, and thus they were lim-

ited to one. These are 13 (non-planned extraction), 15

(aquifers degradation (basins)), 16 (decrease in dams

levels), 17 (aquifer recharge decrease), and ﬁnally 18

(decrease water availability) which is superposed with

concept 13.

With the same map, we started and kept the

social drivers 3 (population growth), 4 (agriculture

development), 5 (urbanization), and 6 (industry

growth) forced, and turned off C

(Figure 4 Bottom).

After six iterations we obtained:

= [0 0 1 1 1 1 1 1 1 0.6 0 1 1 0.78 1 1 0.6 1 0.9

Concepts that depend of C

remained off. Con-

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Figure 4: Upper: Evolution of the concepts erosion in-

crease(12), non-planned extraction (13), basins degradation

(Cutzamala) (14), aquifers degradation (basins) (15), de-

crease in dams levels (16), aquifer recharge decrease (17),

decrease water availability (18), and social and administra-

tive conﬂicts increase (19) after 6 iterations. Keeping forced

denoted as 1. Bottom: Evolution of the concepts 12, 13,

14, 15, 16, 17, 18 and 19. Turning on the social forcers 3,

4, 5, and 6, and turning off C

cepts 7, 8 and 9, which depend directly of activated

concepts, went to one at the ﬁrst iterations. Con-

cept 10 was activated constantly by Concept 7. Con-

cept 11 remained off. Concepts 12 (erosion increase)

and 13 (non-planned extraction) went to one. Con-

cept 14 (basins degradation Cutzamala) stabilized at

0.78. Concepts 15 (aquifers degradation (Basins))

and 16 (decrease in dams levels) went quickly to 1.

Concept 17 (aquifer recharge decrease) stabilized in

0.6. While Concept 18 (decrease in water availabil-

ity) went to 1 after 3 iterations. And ﬁnally, Con-

cept 19 (social and administrative conﬂicts increase)

remained as 0.9 because it is activated by Concept 18.

The substantial difference between forcing the sys-

tem with social or climatic factors is shown in nodes

12 (erosion Increase) and in 14 (basins degradation,

Cutzamala).

3.6 Min - Max Criteria for the Analysis

of Inﬂuences over the Decrease in

Water Availability

As we said in the introduction, we can use the

min - max criteria to analyze how much a node

causes another. In this case, we wanted to compare

the inﬂuence of climatic and social factors over

the decrease in water availability. To estimate the

total effect of temperature increase over decrease in

water availability, we had to consider the effective

paths: I

(1, 9, 13, 16, 18) = min{M, H, H, H, H} = M,

(1, 9, 13, 15, 17, 18) = min{M, H, H, M, M} =

M, I

(1, 20, 16, 18) = min{H, H, H} = H,

(1, 20, 17, 18) = min{H, H, M} = M,

(1, 2, 16, 18) = min{M, H, H} = M,

(1, 2, 17, 18) = min{M, H, H} = M,

(1, 11, 12, 14, 16, 18) = min{L, H, M, H, H} = L.

The total effect is the maximum of the indirect

effects, in this case equal to H. But if we analyze

closely the only path which has indirect effect H is

(1, 20, 16, 18) = min{H, H, H} = H, so if we want

to change the effect of the temperature increase over

decrease in water availability we need to focus on

the processes among: temperature increase → evap-

oration increase. evaporation increase → decrease

in dams levels. decrease in dams levels → decrease

water availability. These three edges are practically

out of our control, but we can infer that if we want to

counteract the inﬂuence of temperature increase over

the system, we must probably focus on strategies to

avoid the decrease in dam levels.

On the other hand, when we analyzed the so-

cial drivers (as a whole), we found that the total

effect of all of them is High, but only one Indi-

rect effect which is ”High” is I

3→18

(3, 9, 13, 16, 18) =

min{H, H, H, H}. Like we just discussed, this obser-

vation can also help us for decision making and strate-

gic planning.

4 CONCLUSIONS

We could observe, from the feedback processes form

section 3.1, that both social and climatic drivers will

lead the system to an undesirable state. The only dif-

ference between the two feedbacks of social and cli-

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501

matic drivers are nodes 12 (erosion increase) and 14

(basins degradation, Cutzamala). Both of them are

turned on due to social drivers.

The results found in section 3.2 reconﬁrm that so-

cial drivers have a high inﬂuence over the network.

By analyzing the hidden patterns obtained by keep-

ing the different drivers ON, the difference were in

the following nodes: waste waters discharge increase

(7), vegetal cover loss (8), degradation in the water

quality (10), erosion increase (12) and basins degra-

dation, Cutzamala (14). That were activated by the

social drivers and not by the climatic drivers.

In section 3.3 the system was forced under two

different climate change scenarios. The considera-

tions made were the decrease and the increase in PCP.

The resulting vector in comparison with the vector v

obtained in the case (a) in section 3.2,(base scenario,

that we obtained when nodes 1 and 2 were forced in

section 3.2), differed only on the nodes 11, 12 and

14, and there was no difference when we considered

a PCP decrease or increase. This is a consequence of

the particular structure of the system.

When considering weighted edges for the whole

system (section 3.5) we validated the results obtained

in previous sections. Since the difference between

social and climatic factors was in nodes 12 (erosion

increase) and in 14 (basins degradation, Cuzamala)

once again. In section 3.4 we could identify that, even

though the system behavior was sensible to changes in

the weights, it maintained the general tendency.

In section 3.6 we found that the “causality” that

climate and social drivers (as a whole) imparted over

the decrease in water availability was ”high”. But in

both cases the “high” causality is centered on a par-

ticular processes.

From these results we can conclude, ﬁrst of all,

that the system is signiﬁcantly affected by climatic an

social drivers, i.e. both can trigger the system and lead

it into an undesirable state. Moreover, it appears that

the strength of social drivers are greater than those of

the climatic drivers. Since the social drivers in Mex-

ico city are currently on, climatic drivers will act as an

accelerator for the degradation processes on the water

system. Therefore both drivers should be taken into

account for policies design.

The general recommendations after the analysis

are focused on two branches:

Climatic drivers:

1→18

(1, 20, 16, 18) = min{H, H, H} = H.

Social drivers:

3→18

(3, 9, 13, 16, 18) = min{H, H, H, H}.

Which have a “high” inﬂuence over water availabil-

ity. Further investigation is needed for each node and

edge on the two branches, to design policies and so-

lutions.

A general conclusion based on this FCM for wa-

ter availability in Mexico city, is that the degradation

process will occur, given the present conditions, with

climatic drivers or without them, as social drivers are

more inﬂuential on the network. The social situation

that operates over the water system is pushing the sys-

tem into an undesirable state, and only with a more in

depth study of the interactions among the nodes we

will know whether or not the system can return to its

equilibrium state.

It is remarkable that the system’s dynamic did not

change wether a consideration of an increase or a de-

crease in precipitation was made. It is also notable

how the system is sensible to changes in the weight of

the edges but without changing its general tendency.

ACKNOWLEDGEMENTS

The present work was developed with the support of

the Programa de Investigaci

on en Cambio Clim

atico

(PINCC) of the Universidad Nacional Aut

onoma de

exico (UNAM).

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