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Summary of expected outcome, concepts activated and iteration process for three sample cases. This is a highly likely outcome given that those with higher incomes and education are better able to identify and seek help for their mental illnesses which increases the likelihood that they will avoid incarceration. However, the strength of income and education as protective factors against increasing mental illness is shown to be ineffective and the level of mental illness continues to rise.

Despite the increase in mental illness, education and income will ensure an ongoing ability to provide shelter, resulting in homelessness being an extremely unlikely outcome. However, due to the known cumulative negative effects of addiction, social isolation increases, signalling the likelihood that, over time, there will be an increased possibility of family breakdown and greater challenges controlling the addiction resulting in the increased likelihood of crime.

Iteration 2 demonstrates the actions of all the concepts present in iteration 1 continuing to exert force on the model with the addition of an increase in mental illness caused by the ongoing addiction resulting in an increasing likelihood of homelessness. As the model continues to iterate, the addictions contribute to increasing social isolation and criminal behavior resulting in a greater likelihood of family breakdown.

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At this point the protective factors of education, income and counselling are overwhelmed by the ongoing addictions and resulting mental illness and crime and the likelihood of homelessness rises. However, given that education and income continue to exert force, homelessness is not a certainty. Given the fully explainable results of the model and the fact that it was able to achieve stability after iterating, it was determined that the model functioned properly, and the process of refining the concepts through the search of timely empirical literature was conducted.

To refine the edge weights on the FCM, timely, empirical literature was searched. The original causal map was referred to for the paired concepts such as, education and homelessness. These linked terms were then searched using the academic search engine, Google Scholar. Numerous articles were retrieved and scanned for each pair of linked concepts using only recently published since the year , peer reviewed, empirical articles.

Linguistic statements were required to be in the antecedent - consequent form as earlier described.

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To maintain the semantic consistency amongst various concepts, Oxford Canadian Dictionary [ 10 ] was followed. To calculate the quantitative weight values for each edge, first the qualitative weight values for each of the retrieved linguistic terms was assessed. A Likert-type scale was devised to determine the qualitative weight of each linguistic term. We only consider five qualitative values for the sake of simplicity. However, the scale could be less or more than five, depending on the intricacies of the system under consideration.

Consensus on meaning was achieved through discussion and vote.

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This process resulted in a scale of ordered and ranked values for each concept pair. In the case of disagreement or uncertainty regarding the precise meaning of the words, Oxford Dictionary Online was referenced for definitions and synonyms. Once the different qualitative weight values were determined for each linguistic term, they were then collected into their groups of three and applied to the revised FCM. Following the application of the qualitative values to the FCM the values were then converted to quantitative weight values using FL theory.

Each link was first expressed as a fuzzy rule then used in the Fuzzy Inference System FIS to generate a crisp numeric value. This graded statement would then be transformed using the rule statement:. The linguistic term ON is a binary variable. ON denotes the presence of the concept and VH denotes the weight value qualitatively. For simplicity sake, triangular membership functions have been used as suggested in [ 85 ].

Interested readers can find more detailed explanation on membership functions in [ 86 ]. This means that an increase in addiction in a society will lead to an increase in levels of homelessness. All three studies indicated that as levels of addiction increase they exert a positive effect resulting in increases in levels of homelessness. This means that with higher levels of education in a society there will be lower levels of homelessness.

Therefore, the impact of education on homelessness is modeled as negative - increases in education lead to decreases in homelessness. All literature scanned indicates that as education rises, homelessness falls. The first study stated that the impact of education on homelessness was low , the second, medium , and the third, high. For each edge, we constructed an individual FIS and the defuzzified value, in this case 0.

More information about the procedure can be found in [ 87 - 90 ]. Experimentation with the weighted FCM was conducted, see Algorithm 1 , to ensure that it would perform as expected and that the map had captured the dynamics of the factors which affect levels of homelessness. This choice is made as we are interested in understanding the impact of increase or decrease of initial concept values on the overall stability of the map [ 91 ]. Each example case had a variety of concepts activated at varying levels.

The models were then permitted to iterate as necessary to reach a stable state no further movement, positive or negative, for all concepts in the model. Final iterations are reported for each model. It has been empirically determined that these concepts are often found together and often precede homelessness [ 52 , 70 , 83 ]. Addiction and mental illness are often co-morbid and both commonly precede family breakdown [ 51 ].


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During times of increased addiction and mental illness in society it is the usual reaction of the government to put into place policies and funding which will address these problems [ 93 ]. Activated concepts at levels most likely to result in homelessness with graphical representation of impact of concepts on levels of homelessness over time. In this case, the protective factors of education, income, and social network support protect society from the negative effects of addiction and prevent homelessness.

The link between higher levels of education and higher levels of income have been well documented [ 72 ].

Given that education prepares individuals to think creatively and to problem-solve, it is surmised that those with higher levels of education would have a greater ability to negotiate the complex rules that often are associated with government assistance. Those who are wealthy and educated are also much more likely to be capable of identifying and acquiring the services they might need, such as being able to pay for family counseling rather than being wait-listed for government supplied family counseling.

Over time, the threat of family breakdown is also eliminated and income and education both rise back to their initial levels. Activated concepts at levels least likely to result in homelessness with graphical representation of impact of concepts on levels of homelessness over time. Though addictions are shown as present in this modeled society, the low levels are unable to overwhelm the model.

Through model testing it became apparent that levels of addiction lower than 0.

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Much of the empirical literature support this [ 41 , 59 , 78 ]. Those with high levels of social support such as family, church, social groups, community groups, school friends and community friends are often better able to weather threats such as addictions and family breakdown. In this case, the protective factors of education and income delay the onset of homelessness but are insufficiently strong to prevent rising levels as the model iterates. Over time, due to family breakdown and the diminishing social network support, addictions begin to rise and as addictions rise, the likelihood of homelessness rapidly increases.

This model demonstrates, once again, the importance of family and social support as well as the incredibly negative effects of drug addiction, both as a cause and result of family breakdown. This allowed for confidence that the model was functioning as it ought to and that we had captured not only a number of the integral aspects which contribute to homelessness, but that they were functioning in the direction and strengths which approximated real-life conditions.

Activated concepts at levels most closely representing a typical real-world case with graphical representation of the impact of concepts on levels of homelessness over time. The purpose of this network analysis is to compare the degree of impact each of the concepts exerts on the model. During network analysis, we varied the initial value of a single concept from 0. After several iterations, the value of homelessness was recorded. Then, for each factor, a plot of the value of homelessness versus the initial value of the concept was recorded.

Ideally, for a factor with a positive effect on homelessness, the value of homelessness should increase as the value of the factor increases, gradually converging to a positive value. Concepts which have the reverse - a negative effect on homelessness, should demonstrate a decrease in homelessness as they are increased. Concepts which have higher convergent rates should demonstrate a greater impact on levels of homelessness.

To conduct the network analysis we first set the initial values for all concepts at a level of 0. This told us that the initial value of the factor 0. After analyzing the map, we tried reducing the level of the initial values for all concepts as well as reducing the number of iterations. Through a gradual reduction process we found that by setting the initial concept values at 0. Plots can be examined in pairs or groupings so that the effect of the concepts on levels of homelessness can be compared for both intensity and speed.

Therefore it can be concluded that addictions have a greater impact on homelessness than does cost of housing. Another approach to analyze the most influential factor is through measures of centrality. There are also other measurements for analyzing an FCM, but here we focus on this property.


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  • In this subsection, we describe the results of the analysis based on two types of centrality: degree centrality and closeness centrality. Degree centrality of a graph indicates how strongly a concept node in a FCM affects other concept nodes of the graph [ 96 ]. Closeness centrality of a node is the inverse of the sum of the lengths of the shortest paths between that node and all other nodes.

    Closeness centrality indicates how quickly a concept node affects other nodes of the FCM [ 96 ]. Note : For closeness centrality the distance measured between each pair of nodes is the inverse of the weight of the corresponding edge in the FCM. If there is no edge between nodes then the distance from the one node to the second node would obviously be infinite. Since the FCM is not strongly connected, the length of the shortest path for some pairs of nodes is, in fact, infinite.

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    This then causes the closeness centrality for that node to drop to zero. This makes the centrality of all nodes to be zero. To conquer this problem, we choose a numerical value which is large enough to be considered as an infinite value. Since the distance measure between each pair of nodes is defined as the inverse of the weight between the nodes of the FCM, the greatest distance between each two nodes would be 4. Regarding this value, we picked as an large enough value.

    This approach is similar to the Big-M method described in operation research theories [ 97 ]. These results are consistent with the results of the overall experiment. This study demonstrates the efficacy of using FCM to graphically represent and simulate the actions and interactions present in the social, personal, and structural factors related to homelessness. The FCM is particularly suited to modeling this type of problem due to its ability to incorporate vast amounts of information, synthesizing what is known about a problem and then allowing for meaningful simulations.

    The FCM is particularly suitable due to its dynamic nature and ability to simulate potential policy changes and show predicted outcomes on levels of homelessness. The problem of homelessness is really situated in factors that occur at the micro-, meso-, and macro-levels of society; future research should aim to refine the FCM by sorting factors into their appropriate levels thereby allowing differentiation between what the individual is potentially capable of controlling and that which he or she is not.

    This would allow for clearer identification of where government policy changes would have the greatest effect. Future refinements must also capture the effect of time. Many factors affect the system differently as time progresses i.