Comments on Six Degrees of Separation based on the le Pool and Kochen Modelsgendary
In this article we discuss six degrees of separation, which has been suggested by Milgram’s famous experiment,, from a theoretical point of view again. Though Milgram’s experiment was partly inspired to Pool and Kochen’s study  that was made from a theoretical point of view. At the time numerically detailed study could not be made because computers and important concepts, such as the clustering coefficient, needed for a network analysis nowadays, have not yet developed. In this article we devote deep study to the six degrees of separation based on some models proposed by Pool and Kochen by using a computer, numerically. Moreover we estimate the clustering coefficient along the method developed by us  and extend our analysis of the subject through marrying Pool and Kochen’s models to our method.
keywords: Six Degrees of separation, Small world Network, Propagation Coefficient Model, Clustering Coefficient,
In 1967, Milgram made a great impact on the world by advocating the concept ”six degrees of separation” by a celebrated paper  written based on an social experiment. ”Six degrees of separation” shows that people have a narrow circle of acquaintances. A series of social experiments made by him and his joint researcher suggest that all people in USA are connected through about 6 intermediate acquaintances. Their studies were strongly inspired by Pool and Kochen’s study . At the time, however, numerically detailed study  could not be made because computers and important concepts, such as the clustering coefficient, needed for a network analysis nowadays, have not yet developed sufficiently.
One of the most refined models of six degrees of separation was formulated in work of Watts and Strogatz,. Their framework provided compelling evidence that the small-world phenomenon is pervasive in a range of networks arising in nature and technology, and a fundamental ingredient in the evolution of the World Wide Web. But they do not examine closely Milgram’s original findings by their model:, especially how influence can the clustering coefficient proposed in their paper  have. We have made a study of them in our previous paper  based on a homogeneous hypothesis on networks. As a result, we found that the clustering coefficient has not any decisive effect on the propagation of information on a network and then information easily spread to a lot of people even in the cases with a relatively large clustering coefficient; a person only needs dozens of friends.
In this article we devote deep study to the six degrees of separation based on some models proposed by Pool and Kochen  by using a computer, numerically. Moreover we estimate the clustering coefficient along the method developed by us  and extend our analysis of the subject through marrying Pool and Kochen’s models to our method. As a result, it seems to be difficult that six degrees of separation is realized in the models proposed by Pool and Kochen on the whole.
The plan of this article is as follows. In the next section we argue on the first idea proposed by Pool and Kochen, where they impose a hypothesis on the average number of acquaintances common to individuals. In the section we give the condition that information spreads to about people within the parameters introduced in the model and evaluate how many people can receive information released from one person after steps of transmission of information. We also argue that the results given by Pool and Kochen are unstable. In section 3, we study their improved model. We there investigate the almost same subjects as ones in the section 2. In section 4, we apply the propagation coefficient model proposed by us  to Pool and Kochen to evaluate the clustering coefficient. Then we discuss whether six degrees of separation is feasible in Pool and Kochen’s models. The section 5 is devoted to summary and consideration.
2 Pool and Kochen Model
Some models for patterns of social contacts were described in Pool and Kochen’s paper . They can be broadly classified into two groups: the model with social strata and the ones without social strata in the population considered. We concentrate our discussion on the former cases, which are described in the section ”The number of common acquaintances” of their paper. There mainly two considerations except for trivial notions are, mainly, described.
is the total population in the region considered. A unique characteristics in their model is to introduce the average number of acquaintances common to individuals such as illustrated in Fig.1. They assumed the following relation between and ;
This means that for example the average number of acquaintances common to five people is smaller than the average number common to four by a factor , which is the same proportion as the number of friends shared by four is to the number shared by three. This is between 0 and 1 and should be statistically estimated. It is also assumed that is independent of
FigD1. A schematic illustration of .@@
When is the number of independent acquaintances standing at intervals of link length away from a person A (- generation from A), we can express , which is the probability that acquaintances standing at - generation from the person A are just an acquaintance of another person B chosen randomly from people except for the acquaintance tree starting from A , as
The Fig.2 shows the situation of Pool and Kochen Model. The following recursion relation holds for ,
where is the average number of acquaintances that any person knows (referred as the propagation coefficient in our paper) and satisfies . By solving this recursion relation, we can find the total population that receive information propagated from A during generations;
Paying attention to , we obtain
which leads to a useful relation and the propagation coefficient .
FigD2. A schematic illustration of Pool and Kochen Model.@@
Fig.3 shows the numerical relation of Eq. (4) for . The value of comes from a slightly large population of the USA. From this, in order to obtain a realistic value of for , extremely small values of are needed such as when , and when . We refer to Bernard et al. [8, 9, 10], where they estimate that the average person has a social circle of about 290 people from empirical studies. We infer (a few) from that value.
From these results, evaluating the total propagation population under appropriate values of and , we obtain the following results; for and , and for and . This means that evaluated is insufficient for information even to spread to only one percent of the total population even in the case with . Since rapidly converges to a constant when generation grows into about in the case of , such as shown in Fig. 4, grows larger in proportion to generation number for . This is a reason that information can not spread over a large population.
Fig.3 - plot for .@@ Fig.4 - plot for , and , .
Fig.5. fig.5a. - plot fig.5b - plot with the interval of
@ Poor and Kochen mainly gave analyses of the relation between and . Analyzing numerically it in some detail, we find that the relation so unstable that it is difficult to get reliable claims. As shown in Fig.5, rapidly grows around and that is to say, rapidly grows for small changing of . Though the relation between and is unreliable around there, the relation between and is stable as shown in Fig.3. Thus the estimation of calculated above is also reliable.
3 Version up Model
As the second step, Poor and Kochen developed their model more fully. There they introduce a set of A’s circle of acquaintances and its complement. denote the individuals in the set . The following assumption are made on the conditional probability ;
where B is a person randomly chosen and the constant should be statistically estimated. Thus we get 
so we have
From these equations, we get 
By using these relations, Poor and Kochen mainly studied about , but did not give no consideration to . In order to do it we only need to solve the recursion relation (9). In this article we numerically estimate for , changing values of and ,. The results are partly given by Fig.6. As expected, increases as becomes larger. We also find a natural result that the more larger is, the smaller is. It is, however, impossible that information does spread to most of the total population even after 10-th generation in the both cases of Fig.6.
Fig.6. Total population for (left) and generation (right) in -.
4 Poor-Kochen Model and Clustering Coefficient
In this section we calculate the clustering coefficients in Poor-Kochen Model
according to the general method developed in the propagation model .
First of all, we tidy the notations used in the model.
is the number of nodes in - generation .
is the number of total nodes or the size of a network.
is the clustering coefficient of a network.
is the contribution to the clustering coefficient produced in .
is the number of edges connected between the same generation .
the number of edges from a node in to nodes of .
is the average of over all nodes relevant to the generation;
Using these quantities, we can express the average degree of a network.
In the propagation model we assumed that was constant, but we now should assume that the population in each generation is constant in Pool-Kochen model. So assuming that the recursion relation on satisfies
where denotes the probability that a node has two acquaintances in the generation earlier than the node. A schematic diagram of the propagation model including the parameter is given by Fig.7. Substituting these expressions into the following equation (16) given by us
When varying , the behavior of in the propagation model is shown in Fig. 8 where and are taken. This value will be proper for inhomogeneous networks with respect to degree distribution , which is actually assumed in this section. For larger values of and such as and , we have checked that increases more rapidly as grows larger. Thus the clustering coefficient does not grow large unless considerably becomes large. It is thought to be difficult that Pool-Kochen model can realize any small worlds from the perspective of this analysis, too. So far networks based on Pool-Kochen models do display a large world property and small clustering coefficient is preferable.
Fig 7. A schematic diagram of the propagation model.
Fig 8. - plot for and .
Pool and Kochen have made some discussions on models with social strata. Since it is, however, though that the models do not bring any correct results, we abandon the pursuit of the models.
5 Summary and Consideration
In this article we numerically analyzed how six degrees of separation can be realized in human networks based on a series of Pool-Kochen models. Moreover we estimate the clustering coefficient of Pool-Kochen models according to the propagation model and explored the possibility of small-worldness.
In result, we found that it is difficult that Pool-Kochen models realize six degrees of separation and also achieve a large clustering coefficient. Recently Kleinfield has fanned some critical discussions to Milgram’s empirical evidence for six degrees of separation . Later Watts ea al. have conducted by far the largest ever small-world experiment by using E-mail, involving 60 thousand E-mail users with targets over 13 countries [12, 13]. As they recognize, their experiment has a positive bias in the choice of E-mail users. The small world problem, however, remains as fascinating psychological mysteries. In my opinion, the meaning of Pool-Kochen’s study is that the models will not do much a better understanding of six degrees of separation but inspired Milgram and so on to study interesting subjects such as six degrees of separation.
Once we discussed it based on homogeneous hypothesis in the propagation model  and find that six degrees of separation likely to materialize somewhat. The hypothesis is, however, no correct. There is considerably deviation in the degree distribution, the clustering coefficient and so on in real networks. To understand six degrees of separation more really, we should introduce some correct distributions into the degree distribution, the clustering coefficient and so on. Newman  has made discussions on it by considering some distribution in the degree distribution and furthermore ”mutuality” which is a quantity that reflects the density of squares in human relations. This direction of study seems to play an important role in the studies of six degrees of separation. The detail research toward the line of this should be made more properly.
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