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?dBBXXX>>>>>>>$p@dB>l>7XXUB777"XX>>7R77<C>|VT>X,-v0>4 COHESIVE SUBGROUPS
Embedded within a network there are often groups of actors who interact with each other to such an extent that they could be considered to be a separate entity. In a friendship network this could be a group of close friends who all socialise together, alternatively in a work environment a collection of colleagues who all support the same football team, or in a network of interacting organizations a collection of organizations which behave as a single unit (sometimes referred to as virtual organizations). We call any such group a cohesive subgroup. In these examples we have identified the underlying cohesive theme which unites the group, this would not necessarily be apparent from the network under study. In examining network data we would first try and detect the cohesive subgroups and then, by looking at common attributes, see if there was some underlying principle that could explain why they identify with each other.
At first sight it may appear easy to identify cohesive subgroups in a network by simply looking at it. Unfortunately it is very easy to miss group members or even whole groups when trying to find cohesive subgroups by simply looking at a network. The position of actors on the page and the preponderance of edges make this task almost impossible to do by hand and we need to resort to algorithms and computers to perform the task for us. This is particularly true if the data was collected either electronically or by questionnaire, but even with observational data it is recommended that a proper and full analysis be undertaken. It should be remembered that some cohesive subgroups are open and want to be identified, but for others there is a strong dis-benefit in identification (For example a cartel, or a drugs ring). It is therefore necessary to have some formal definitions that capture exactly what a cohesive subgroup is. Within the social sciences the notion of a social group is often used casually. It is assumed that the reader has an intuitive grasp of the concept involved and that it is not necessary to present an exact definition. Clearly such an approach cannot be used to analyse real data and we are forced to define precisely what is meant by a cohesive subgroup. There are a large number of possible realisations of the social group concept but we shall only concern ourselves with the more commonly used practical techniques.
4.1 Intuitive Notions
We start by considering the most extreme case of a cohesive subgroup. In this case we expect members of the group to have strong connections with every other member of the group. If we have binary symmetric data, that is data that is represented by a graph, then this would imply that every actor in the group is connected to every other group member. In addition the group has no connections to individuals on the outside. In graph theoretic terms this group would consist of a component of the network which was a complete graph. Clearly such a notion has no place in any practical data analysis as such structures are so rare as to render the concept useless. However this strong intuitive notion allows us to construct models of cohesive subgroups based upon different relaxations of this ideal.
One of the first considerations is the type of data we are dealing with. Our ideal situation involved us looking at symmetric binary data, how do we deal with directed or valued data? Let us first consider non-symmetric data. If the data is binary, then since each pair of actors within the group have a strong connection between them we would expect all ties to be reciprocated. It is therefore not necessary to consider directed networks and so we restrict our attention to undirected relations. If the original data is directed then as a pre-processing stage we should symmetrize it. Since we expect strength within the group we should symmetrize by constructing a network of reciprocated ties only. If there are very few or no reciprocated ties then we could use the underlying graph and simply ignore the directions of the edges. But this clearly is second best and care should be exercised in interpreting any subgroups found under these circumstances. These ideas are still applicable when we consider valued data and consequently we only consider symmetric valued data and apply the same principles in our symmetrization as in the binary case. For our valued data we expect strong ties within the group and weak ties outside.
4.2 Cliques
If we do have undirected binary data then we can relax the condition on our extreme cohesive subgraph by removing the constraint that there are no external links. If in addition we insist that all possible members of the group have been included then we call the resultant structure a clique. A clique is therefore a subset of actors in which every actor is adjacent to every other actor in the subset and it is impossible to add any more actors to the clique without violating this condition. A subgraph in which every actor is connected to every other is called a complete or dense subgraph. We can therefore define a clique as a maximal dense subgraph. Here the term maximal means that we cannot increase its size and still have a dense subgraph. In applications we usually insist that any clique has at least 3 actors and we often increase the minimum size to try and identify the stronger groupings in the data.
We can illustrate the idea of a clique by examining the network in Figure 4.1. We see that nodes 1,2,3 and 4 are all connected to each other, in addition we cannot increase this group and still retain this property. Node 5 is connected to 3 and 4 but not to 1 and 2. It follows that {1,2,3,4} is a clique. Other cliques are {3,4,5}, {7,9,10}
FIGURE 4.1
and {7,8,10}. Note that {1,2,3} is not a clique because it is not maximal (we can add 4 to it). Clearly cliques can overlap so that individual actors can be in more than one clique. In our example we see that nodes 3,4,7 and 10 are all in two cliques. Finally there can be actors who are not in any cliques. Again returning to our example we can see that node 6 is not in any cliques.
An example. Roethlisberger and Dickson (1939) observed data on 14 Western Electric employees working in a bank wiring room. The employees worked in a single room and included two inspectors (I1and I3), three solderers (S1, S2 and S4) and nine wireman (W1 to W9). One of the observations was participation in horseplay and the adjacency matrix for this is
1 2 3 4 5 6 7 8 9 10 11 12 13 14
I1 I3 W1 W2 W3 W4 W5 W6 W7 W8 W9 S1 S2 S4
-- -- -- -- -- -- -- -- -- -- -- -- -- --
1 I1 0 0 1 1 1 1 0 0 0 0 0 0 0 0
2 I3 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 W1 1 0 0 1 1 1 1 0 0 0 0 1 0 0
4 W2 1 0 1 0 1 1 0 0 0 0 0 1 0 0
5 W3 1 0 1 1 0 1 1 0 0 0 0 1 0 0
6 W4 1 0 1 1 1 0 1 0 0 0 0 1 0 0
7 W5 0 0 1 0 1 1 0 0 1 0 0 1 0 0
8 W6 0 0 0 0 0 0 0 0 1 1 1 0 0 0
9 W7 0 0 0 0 0 0 1 1 0 1 1 0 0 1
10 W8 0 0 0 0 0 0 0 1 1 0 1 0 0 1
11 W9 0 0 0 0 0 0 0 1 1 1 0 0 0 1
12 S1 0 0 1 1 1 1 1 0 0 0 0 0 0 0
13 S2 0 0 0 0 0 0 0 0 0 0 0 0 0 0
14 S4 0 0 0 0 0 0 0 0 1 1 1 0 0 0
The program UCINET was used to produce the following 5 cliques.
I1 W1 W2 W3 W4
W1 W2 W3 W4 S1
W1 W3 W4 W5 S1
W6 W7 W8 W9
W7 W8 W9 S4
We note that although these cliques overlap there are two distinct groups, namely {I1,W1,W2,W3,W4,W5,S1} and {W6,W7,W8,W9,S4} together with two outsiders I3 and S2. These two groupings are in exact agreement with the findings of Rothlisberger and Dickson who identified the groups as those at the front of the room and those at the back. In this instance a simple clique analysis has been successful in identifying important structural properties of the network. Unfortunately most analyses are not as straight forward as this one. Two problems can occur. The first is that there is a large number of overlapping cliques and it is difficult to deduce anything from the data. The second problem is that too few cliques are found and so that no important subgroups are identified. We shall return to the first of these problems later in the chapter. The second problem of too few cliques can be addressed by having less stringent conditions for a subgroup to be a cohesive subset.
4.3 k-plexes
The condition for being a clique can be quite demanding. The fact that every member must be connected to every other member without exception is a strong requirement particularly if the group is large. It also means that the data has no room for error, a missing connection, for whatever reason, immediately stops a particular group from being identified as cohesive. The idea of a k-plex is to relax the clique concept and allow for each actor to be connected to all but k of the actors in the group. It follows that in a 1-plex every actor is connected to all other actors except for one, but since we normally do not have self loops (if we do have self loops for any cohesive subgroup method they are ignored) this implies that every actor is connected to every other actor and we have a clique. In a 2-plex every actor is connected to all but one of the other actors. We do again insist that the cohesive subgroup is maximal. We can see that in Figure 4.1 {7,8,9,10} is a 2-plex since 7 and 10 are connected to all the other nodes and 8 and 9 are connected to all but one of the other nodes. Another 2-plex is the clique {1,2,3,4} note we cannot include the actor 5 in this 2-plex since it does not have connections to 1 and 2 (note that {1,2,3,4,5} does form a 3-plex). Consider the subset of actors {5,6,7}. Is this a 2-plex? The answer is yes since each actor is connected to all but one (that is just one) of the other actors. Clearly this group does not fit our intuitive notion of a cohesive subgroup. The problem is that the subgroup is too small to allow us to make the relaxation and still retain the characteristics of a cohesive group. We should therefore increase the minimum size of our 2-plex from 3 to 4. This problem can become worse as we increase the value of k. Again in Figure 4.1 the subset {3,4,5,7,8,10} is a 4-plex since every node is adjacent to all but 4 others (that is each node is adjacent to two others). It can be shown that if for a particular value of k the subset is bigger than 2k-2 then the distance between the nodes in the cohesive subgraph is always 1 or 2. This formula works well for k bigger than 4 but does not help us for k=2 or 3. The reason for this is that when the subgroups are small the distance between them must be small, in our {5,6,7} 2-plex all the nodes are a distance of 1 or 2. We can combine both of these findings to produce a recommended minimum size for a value of k as given below.
k Minimum size243547k2k-1
We can now define a k-plex as a maximal subgraph containing at least the number of actors given in the table above with the property that every actor is connected to all but k of the other actors.
Returning to Figure 4.1, the 2-plexes are {1,2,3,4},{1,3,4,5},{2,3,4,5} and {7,8,9,10} and the 3-plexes are {1,2,3,4,5}.
If we now re-examine the wiring room matrix and perform a k-plex analysis then we obtain the following 2-plexes
I1 W1 W2 W3 W4 S1
I1 W1 W3 W4 W5
W1 W2 W3 W4 W5 S1
W6 W7 W8 W9 S4
We can again clearly see the two groups and we have managed to reduce the number of cohesive subgroups as the k-plexes have allowed us to merge together two overlapping cliques. If we increase k and look at the 3-plexes then we obtain
I1 W1 W2 W3 W4 W5 S1
W6 W7 W8 W9 S4
Which are precisely the groups identified by Roethlisberger and Dickson.
4.4 Overlap
As we have already noted cliques and k-plexes may overlap. In fact it could be argued that this is a positive and necessary feature but it can bring with it certain difficulties. When there are a large number of cohesive subgroups the overlap itself can hide features of the clique structure. A graph with just 21 vertices can have up to 2,187 different cliques! Whilst it is true that this is unlikely to occur in any real data it does give an indication of the possible scale of the problem. To see how the problem can manifest itself in real data we consider the Kapferer (1972) tailor shop data. Bruce Kapferer observed interactions in a tailor shop in Zambia (then Northern Rhodesia) at two different times (the time points were seven months apart). One of his data matrices that he labelled 'sociational' contained information about friendship and support. A clique analysis of the later time period for the sociational matrix yields 118 cliques amongst the 39 actors in the study. The cliques are listed in Table 4.1.
1: ABRAHAM HASTINGS CHISOKONE MUKUBWA KALAMBA IBRAHIM MESHAK
2: NKUMBULA ABRAHAM HASTINGS CHISOKONE MUKUBWA MESHAK
3: ABRAHAM NKOLOYA HASTINGS CHISOKONE MUKUBWA MESHAK
4: KAMWEFU NKUMBULA ABRAHAM HASTINGS CHISOKONE MUKUBWA
5: KAMWEFU ABRAHAM HASTINGS CHISOKONE MUKUBWA IBRAHIM
6: HASTINGS CHISOKONE MUKUBWA KALAMBA JOHN
7: HASTINGS CHISOKONE MUKUBWA JOHN CHOBE
8: HASTINGS CHISOKONE MUKUBWA KALAMBA IBRAHIM MESHAK JOSEPH
9: NKOLOYA HASTINGS CHISOKONE MUKUBWA MESHAK JOSEPH
10: MATEO CHISOKONE ENOCH MUKUBWA IBRAHIM
11: KAMWEFU NKUMBULA ABRAHAM ZULU CHISOKONE MUKUBWA
12: KAMWEFU ABRAHAM ZULU CHISOKONE MUKUBWA IBRAHIM
13: ABRAHAM NKOLOYA ZULU CHISOKONE MUKUBWA
14: ABRAHAM ZULU CHISOKONE MUKUBWA KALAMBA IBRAHIM
15: CHISOKONE MUKUBWA KALAMBA IBRAHIM MESHAK JOSEPH HENRY
16: CHISOKONE MUKUBWA KALAMBA JOHN HENRY
17: KAMWEFU NKUMBULA ABRAHAM LYASHI HASTINGS MUKUBWA
18: KAMWEFU ABRAHAM LYASHI HASTINGS MUKUBWA IBRAHIM
19: ABRAHAM NKOLOYA LYASHI HASTINGS MUKUBWA ADRIAN
20: ABRAHAM LYASHI HASTINGS MUKUBWA KALAMBA IBRAHIM
21: KAMWEFU NKUMBULA ABRAHAM LYASHI ZULU MUKUBWA
22: KAMWEFU ABRAHAM LYASHI ZULU MUKUBWA IBRAHIM
23: ABRAHAM NKOLOYA LYASHI ZULU MUKUBWA
24: ABRAHAM LYASHI ZULU MUKUBWA KALAMBA IBRAHIM
25: KAMWEFU NKUMBULA ABRAHAM LYASHI LWANGA MUKUBWA
26: KAMWEFU ABRAHAM LYASHI LWANGA MUKUBWA IBRAHIM
27: ABRAHAM NKOLOYA LYASHI LWANGA MUKUBWA
28: LYASHI HASTINGS MUKUBWA KALAMBA IBRAHIM JOSEPH
29: NKOLOYA LYASHI HASTINGS MUKUBWA JOSEPH
30: LYASHI LWANGA MUKUBWA IBRAHIM JOSEPH
31: NKOLOYA LYASHI LWANGA MUKUBWA JOSEPH
32: NKOLOYA LWANGA MUKUBWA MPUNDU
33: LWANGA ENOCH MUKUBWA MPUNDU
34: LWANGA ENOCH MUKUBWA IBRAHIM
35: ABRAHAM LWANGA MUKUBWA IBRAHIM MESHAK
36: NKUMBULA ABRAHAM LWANGA MUKUBWA MESHAK
37: ABRAHAM NKOLOYA LWANGA MUKUBWA MESHAK
38: NKOLOYA LWANGA MUKUBWA MESHAK JOSEPH
39: LWANGA MUKUBWA IBRAHIM MESHAK JOSEPH
40: NYIRENDA MUKUBWA ZAKEYO JOHN
41: NYIRENDA MUKUBWA KALAMBA JOHN
42: NYIRENDA MUKUBWA JOHN CHOBE
43: NYIRENDA MUKUBWA KALAMBA MESHAK
44: HASTINGS MUKUBWA ZAKEYO ADRIAN
45: HASTINGS MUKUBWA ZAKEYO JOHN
46: MUKUBWA ZAKEYO MPUNDU
47: ABRAHAM NKOLOYA HASTINGS MUKUBWA MESHAK ADRIAN
48: ABRAHAM MUKUBWA MESHAK ADRIAN MUBANGA
49: MUKUBWA MESHAK ADRIAN HENRY MUBANGA
50: HASTINGS MUKUBWA ADRIAN CHOBE
51: MUKUBWA ADRIAN CHOBE MUBANGA
52: MUKUBWA KALAMBA KALUNDWE
53: MUKUBWA KALUNDWE MPUNDU
54: NKOLOYA ZULU MUKUBWA MPUNDU
55: MUKUBWA IBRAHIM MESHAK JOSEPH HENRY MUBANGA
56: ABRAHAM MUKUBWA IBRAHIM MESHAK MUBANGA
57: NKUMBULA ABRAHAM MUKUBWA MESHAK MUBANGA
58: MUKUBWA JOHN HENRY MUBANGA
59: MUKUBWA JOHN CHOBE MUBANGA
60: MUKUBWA JOHN CHOBE MABANGE
61: MUKUBWA MESHAK MABANGE
62: SEAMS MATEO CHISOKONE
63: KAMWEFU ABRAHAM CHIPATA CHILWA LYASHI ZULU IBRAHIM
64: ABRAHAM CHIPATA NKOLOYA CHILWA LYASHI ZULU
65: KAMWEFU NKUMBULA ABRAHAM CHIPATA LYASHI ZULU
66: ABRAHAM CHIPATA NKOLOYA CHIPALO LYASHI HASTINGS
67: KAMWEFU NKUMBULA ABRAHAM CHIPATA LYASHI HASTINGS
68: KAMWEFU ABRAHAM CHIPATA LYASHI HASTINGS IBRAHIM
69: ABRAHAM CHIPATA NKOLOYA CHIPALO HASTINGS MESHAK
70: NKUMBULA ABRAHAM CHIPATA HASTINGS MESHAK
71: ABRAHAM CHIPATA HASTINGS IBRAHIM MESHAK
72: ABRAHAM CHIPATA IBRAHIM MESHAK MUBANGA
73: NKUMBULA ABRAHAM CHIPATA MESHAK MUBANGA
74: CHIPATA ENOCH IBRAHIM
75: CHIPATA LYASHI HASTINGS IBRAHIM JOSEPH
76: CHIPATA NKOLOYA LYASHI HASTINGS JOSEPH
77: CHIPATA HASTINGS IBRAHIM MESHAK JOSEPH
78: CHIPATA NKOLOYA HASTINGS MESHAK JOSEPH
79: CHIPATA IBRAHIM MESHAK JOSEPH MUBANGA
80: ABRAHAM CHILWA LYASHI ZULU KALAMBA IBRAHIM
81: KAMWEFU ABRAHAM CHILWA ZULU CHISOKONE IBRAHIM
82: ABRAHAM CHILWA ZULU CHISOKONE KALAMBA IBRAHIM
83: ABRAHAM NKOLOYA CHILWA ZULU CHISOKONE
84: CHIPALO LYASHI HASTINGS BEN
85: KAMWEFU LYASHI ZULU PAULOS IBRAHIM
86: LYASHI ZULU PAULOS KALAMBA IBRAHIM
87: MATEO ENOCH PAULOS IBRAHIM
88: PAULOS KALAMBA IBRAHIM MESHAK
89: PAULOS KALAMBA KALUNDWE
90: SIGN IBRAHIM CHILUFYA
91: NYIRENDA ZAKEYO BEN JOHN
92: HASTINGS ZAKEYO BEN JOHN
93: ZAKEYO BEN MPUNDU
94: CHISOKONE KALAMBA JOHN WILLIAM
95: CHISOKONE KALAMBA JOSEPH WILLIAM
96: CHISOKONE JOHN WILLIAM CHOBE
97: CHISOKONE WILLIAM CHOBE KALONGA
98: CHISOKONE JOSEPH WILLIAM CHRISTIAN
99: CHISOKONE WILLIAM CHRISTIAN KALONGA
100: LYASHI KALAMBA JOSEPH WILLIAM
101: MPUNDU WILLIAM CHRISTIAN
102: JOHN WILLIAM CHOBE MUBANGA
103: WILLIAM CHOBE MUBANGA KALONGA
104: JOSEPH WILLIAM MUBANGA CHRISTIAN
105: WILLIAM MUBANGA CHRISTIAN KALONGA
106: CHRISTIAN KALONGA MABANGE
107: JOSEPH MUBANGA CHRISTIAN CHILUFYA
108: MUBANGA CHRISTIAN KALONGA CHILUFYA
109: NKUMBULA ZULU CHISOKONE KALONGA
110: CHISOKONE HENRY KALONGA
111: NYIRENDA CHOBE KALONGA
112: NKUMBULA LWANGA KALONGA
113: HENRY MUBANGA KALONGA CHILUFYA
114: NKUMBULA MUBANGA KALONGA
115: CHOBE KALONGA MABANGE
116: IBRAHIM JOSEPH HENRY MUBANGA ANGEL CHILUFYA
117: CHISOKONE IBRAHIM JOSEPH HENRY ANGEL
118: JOHN HENRY MUBANGA CHILUFYA
TABLE 4.1
It is quite clear that it is fairly difficult to deduce any information from the analysis except that there are a lot of cliques that overlap in a complicated fashion. One possible way forward would be to try and reduce the number of cliques by increasing the minimum size allowed. Unfortunately this particular data set has quite a few large groups and whilst this approach could be useful in general it is not the solution here. An alternative strategy would be to try and remove or reduce the overlap by performing some additional analysis on the cliques themselves. We shall now explore this approach in some detail.
We can use the cliques or k-plexes (we shall use the term group to mean either in this section) to obtain a measure of association between each pair of actors. If actor X is in a lot of groups with actor Y it is reasonable to assume that X and Y are reasonably close. In fact we can build a proximity matrix which tells us how many times each pair of actors in our network are in the same group together. We call this matrix the group co-membership matrix A, where A(i,j) = the number of times i is in a group with j. The ith diagonal entry gives the number groups containing actor i. Below we construct the group co-membership for the Games matrix in the Bank wiring room.
1 2 3 4 5 6 7 8 9 10 11 12 13 14
I1 I3 W1 W2 W3 W4 W5 W6 W7 W8 W9 S1 S2 S4
-- -- -- -- -- -- -- -- -- -- -- -- -- --
1 I1 1 0 1 1 1 1 0 0 0 0 0 0 0 0
2 I3 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 W1 1 0 3 2 3 3 1 0 0 0 0 2 0 0
4 W2 1 0 2 2 2 2 0 0 0 0 0 1 0 0
5 W3 1 0 3 2 3 3 1 0 0 0 0 2 0 0
6 W4 1 0 3 2 3 3 1 0 0 0 0 2 0 0
7 W5 0 0 1 0 1 1 1 0 0 0 0 1 0 0
8 W6 0 0 0 0 0 0 0 1 1 1 1 0 0 0
9 W7 0 0 0 0 0 0 0 1 2 2 2 0 0 1
10 W8 0 0 0 0 0 0 0 1 2 2 2 0 0 1
11 W9 0 0 0 0 0 0 0 1 2 2 2 0 0 1
12 S1 0 0 2 1 2 2 1 0 0 0 0 2 0 0
13 S2 0 0 0 0 0 0 0 0 0 0 0 0 0 0
14 S4 0 0 0 0 0 0 0 0 1 1 1 0 0 1
If we look at row 5 column 6 we can see that W3 was in a clique with W4 on 3 occasions, the diagonal entry for row 4 indicates that W2 was in two cliques. These results can be checked by referring back to the list of cliques displayed in section 4.2.
We note that the group co-membership matrix is a proximity matrix in which larger values indicate a stronger link, that is it is a similarity matrix. This matrix can then be submitted to a hierarchical clustering procedure such as the single-link method. The result will be sets of non-overlapping nested clusters of actors. Applying this technique to the group co-membership matrix for the Games data results in the cluster tree diagram given in Figure 4.2. In this diagram we can clearly see the two outsiders
S2 and I3, and at the 1 level the two major groupings {I1 W1 W2 W3 W4 W5 S1} and {W6 W7 W8 W9 S4} identified previously. In addition we can see that W1,W3 and W4 are at the most active in the first group and W7,W8 and W9 are the most active in the second.
In this simple example the method has clearly worked well. But care is needed in interpreting the results of using a technique like this. The clustering is based upon the amount of activity of pairs of actors and not on the strength or overlap of the groups. A consequence of this is that in a network with say one large homogeneous group and another group consisting of a large number of overlapping cliques the analysis will be biased towards the complex overlapping structure. Another possible objection to the method is that it completely eliminates one of the desirable features of cohesive subgroups, namely overlap. Our own experiences tell us that it is quite common for actors to be in more than one group. We shall therefore look at an alternative but related approach that tries to reduce rather than destroy the amount of overlap.
In the method outlined above the approach was to cluster the actors based upon the frequency of pairwise group membership. A dual approach would be to cluster the groups based upon a measure of pairwise overlap. The result would be an hierarchical clustering of groups, each group at a particular level would be uniquely placed in a cluster. But since the actors are not uniquely identified into groups the clusters of groups could consist of actors which belong to more than one cluster. We assume that group X and group Y are similar if they share a lot of actors. We therefore define a similarity matrix called the co-group matrix B where B(i,j) is the number of actors group i has in common with group j. The diagonal entries will give the size of the groups. The following matrix is the co-group matrix for the clique analysis of the Games data. Notice that the matrix is 5x5 since we identified 5 cliques.
1 2 3 4 5
-- -- -- -- --
1 5 4 3 0 0
2 4 5 4 0 0
3 3 4 5 0 0
4 0 0 0 4 3
5 0 0 0 3 4
We see that since the entry in row 2 column 3 is 4 then cliques 2 and 3 have 4 actors in common these are W1,W3,W4 and S1. If this is now submitted to a single link hierarchical clustering procedure we obtain the clustering given in the tree diagram in Figure 4.3. Here we see cliques 1,2 and 3 form a cluster and cliques 4 and 5 form a
separate cluster. The actors in cliques 1,2 and 3 are {I1 W1 W2 W3 W4 W5 S1} and in cliques 4 and 5 are {W6 W7 W8 W9 S4} repeating the analysis obtained from the actor based clustering. The reason these two methods agree is that there is no overlap between the two basic groups. We shall therefore give one further example (in less detail) which demonstrates the differences between the two approaches.
Wayne Zachary (1977) collected data on the members of a university karate club. One of the data matrices he collected records interactions between members of the club outside of club activities. During the study there was a dispute and this resulted in the club splitting into two. The data only considers those members of the club whom joined one of the two new clubs after the split. There were 34 actors involved in the study and a clique analysis reveals 25 overlapping cliques (we do not report the individual cliques here). The cliques were then submitted to the two techniques outlined in this section for dealing with overlap. The results of the clusterings are presented in Figure 4.4. Since there are not many levels compared to the number of actors or cliques we have presented the results as cluster diagrams rather than tree diagrams. Tree diagrams (and dendrograms) are unclear when there are far fewer levels than actors. The top diagram corresponds to the group co-membership matrix
1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 3 1 2 3 3 3 3
Level 0 2 8 3 4 5 6 7 8 9 0 1 2 3 5 6 7 8 9 1 1 2 3 4 5 6 7 9 1 4 0 2 3 4
----- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXX
5 . . . . . . . . . . . . . . . . . . . . XXX . . . . . . . . . . XXX
3 . . . . . . . . . . . . . . . . . . . . XXXXXXX . . . . . . . . XXX
2 . . . . . . . . . . . . . . . . . . . . XXXXXXXXXXXXXXXXXXXXXXXXXXX
1 . . XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
0 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2
Level 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
----- - - - - - - - - - - - - - - - - - - - - - - - - -
4 . . . . . . . . . . . XXX . . . . . . . . . . . .
2 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX XXXXXXXXX .
1 XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
(and is therefore a clustering on the 34 actors) and the bottom diagram to the co-group matrix (on the 25 cliques).We shall first consider the group co-membership matrix. There are no clear distinct groupings but we do have some information on the activities of individuals. Actors 33 and 34 are in 8 different cliques together and this is indicates that this pair are important within the group and are possibly taking on some kind of leadership role. (We could use centrality to measure this, see chapter 6). Actors 1 and 2 are similarly highly active and we could draw the same conclusions. Looking at the lower level we see that the overlap between pairs of actors causes the groups to merge. One possible explanation is that there are two basic groups one led by actor 1 or 2 the other by actor 33 or 34 with the rest of the club surrounding each group to some extent but still with a high degree of integration. If this is the case then we would have two core groups with an associated periphery but the periphery of each group interacts with the other.
Now consider the co-group clustering. Clearly at level 2 we have two main clusters namely 1 through 19 and 20 through 24 with cluster 25 as an outsider. The first cluster has a core consisting of 12 and 13 the second group is more uniform, possibly because it is smaller. The cliques 12 and 13 contain the actors 1,2,3,4,8 and 14 and we would expect the leaders of this group to be among these actors. The cliques 20 through 24 contain actors 1,5,6,7,11 and 17; the fact that actor 1 occurs in both groups is strongly indicative of a leadership role and indicates that these two groups are probably factions within a larger group. The clique 25 contains actors 25,26 and 32 and we would expect these to be members of a different grouping.
In performing an analysis we should not view these techniques as alternatives but as of different views of the same data. They are both, after all, simply a secondary analysis of the clique structure. In summary we have a strong indication that there are two basic groups one of which is led by actor 1 the other by actor 33 or 34. The core of the first group probably consists of two factions {1,2,3,4,8,14} and {1,5,6,7,11,17} which are united around actor 1. The second group contains actors {25,26,32} but its structure is not clear, this is probably because the peripheral actors all overlap and shroud the more detailed structure. We can try and identify the core groups by increasing the minimum clique size. This has the effect of making clique membership more difficult and therefore removing some of the 'noise' from the data. Increasing the minimum clique size to 4 and repeating the overlap analysis reveals two distinct groups within the data (both overlap techniques produce the same answer). We find {1,2,3,4,8,14} (as before) and {9,24,30,31,33,34} which we believe to be the core of the second group. The ethnographic description given by Zachary prior to the split in the club identifies two groups one around actor 1 the other around actor 34. Group 1 has as its core members {1,2,3,4,5,6,7,8,11,12} with {13,14,18,20,22} less strongly attached and Group 2 has as its core members {21,23,30,31,32,33,34} with {9,16,24,25} less strongly attached, actors 10,17 and 19 were not placed in either group. All our results are consistent with this description. Clearly were able to deduce that there were two basic groups but not able to place all the actors into one of the groups. We shall now turn our attention to techniques that achieve precisely this.
4.5 Factions
In this section we shall look at a class of methods that partition the whole population into a pre-determined number of groups. The first important feature of the techniques we shall be introducing is that they form a partition, this means that every actor must be placed into a unique group. When we determined cliques and k-plexes we allowed actors to be placed in more than one group as well as accepting that some actors did not belong to any group. We now insist that every single actor in the network is assigned to one and only one group. Furthermore, when applying these methods we shall have to determine the number of groups a priori. This is because the algorithm we shall be using tries to fit the actors into the groups and then measure how well it has done.
We shall illustrate the ideas by a simple example. We first need some measure of how well a partition has divided the actors into cohesive subgroups. One way would be to calculate the density of each of the groups and then sum them. If we wish to partition our data into three groups then the best value we could achieve using this measure is 3. This would occur when each group had the maximum density. Any measure used in this fashion is more commonly referred to as a cost function and the value of the function for a particular partition is called the fit.
Our algorithm proceeds as follows. First arbitrarily assign everyone to one of the groups and then calculate the fit to see how good the partition is. Next move some (often just one) actors from one group to another calculate the fit again and see if there is any improvement. Continue in this way until no more improvement is possible and the result is the required partition.
Initial configuration Fit = 1.6
Alternative 1: move 4 to group 1, Fit = 1.67
Alternative 2: move 2 to group 2, Fit = 1.53
FIGURE 4.5
By way of example let us consider partitioning the network shown in Figure 4.5 into two groups. The diagram gives an initial partition, if we use a simple density measure as the cost function then this gives a fit of 1.6. We now consider two alternative
moves, the first is to move actor 4 into group 1 and the second is to move actor 2 into group 2. Both alternatives are displayed in Figure 4.5 and the resultant fits are 1.67 for the first move and 1.53 for the second. Since 1.67 is higher than the original value of 1.6 we accept this move and keep this assignment as our best fit so far. We can continue to improve on this by starting from this new configuration and then moving 5 into group 1 to give a fit of 1.7. In fact this new grouping is the best possible and would represent the partition required.
At first sight it would seem the procedure is simple. Start from a random partition, move actors around to find the best improvement and use this as the starting point for a new set of moves. Continue this process until no more improvement can be made and the result is the best partition. Unfortunately this method (called steepest ascent) does not work in most cases for the kind of problems we are concerned with. The reason is that we have started from a given partition and only made small changes to the original. The method is very dependent on the starting position. It is rather like trying to find the highest point in a country by using the following method. Take a map of the country, with a pin randomly pick a starting point. From this point travel up the steepest hill and continue going up until you reach the top. The chances of this being the highest point are very remote, you will probably end up at the top of the nearest hill to the starting point. Exactly the same thing happens with the method outlined above. A more sophisticated search is required. There are three general purpose approaches namely; Simulated Annealing, Tabu Search and Genetic Algorithms. We do not intend to go into these heuristic methods here. Each has its own strengths and weaknesses but computationally they are all similar in terms of their efficiency. In essence each method, in its own way, looks further afield than simply local changes. It does this by evaluating directions that may not at first sight seem very promising. Once this global search has been completed the methods then concentrate on optimising the solution in the way outlined in the above example. It should be noted that this is a deliberately vague description of how the methods work and the interested reader would be advised to look at the specialist literature on these techniques. The general techniques described are known as combinatorial optimisation methods and the application of these algorithms to determining cohesive groups results in a division of the actors into factions.
When applying any of these algorithms to a real problem the user needs to be aware of the possible pitfalls in interpreting the results. The most important factor to be remembered is that there will always be a solution. That is, if the data contains no groups it will still be partitioned into the correct number of classes because the method reports what it considers to be the best of the possible solutions it has looked at. The fact that none of the solutions is any good is ignored. As an extreme example consider a network in which everyone is connected to everyone else. If we selected three groups then any partition into three would have the same fit as any other. The algorithm would therefore retain the original arbitrary split and this would be reported as the partition. Clearly there is only one group within the data and yet the algorithm would report three factions as requested. Note further that the density cost function proposed above would be at the maximum fit of 3. This could be fixed by having a more sophisticated cost function but the underlying problem of always finding a solution still remains. Linked to this is the problem of the uniqueness of the solution. That is there may be more than one partition into factions with the same maximum fit. Consider two complete networks of the same size and then one actor linked to all the actors in the network. Figure 4.6 is such a network in which each of the complete networks consists of three actors. Suppose now we wish to search for two factions, then the two complete graphs will each be in a separate faction but in which group does the actor that is connected to everyone belong? The density fit will be perfect whichever group is selected. There are therefore two possible partitions both of which are valid. In more complex data there may be a number of very different partitions all of which have the same or very similar fit values. One way to check the robustness of a partition is to repeat the analysis a number of times to see whether the final factions are the same or similar.
The simple density cost function used to illustrate the principles in factions is often not effective. The reason is that it only captures part of the intuitive notion discussed in section 4.1. Since we are trying to partition the whole network we are able to include both the idea of internal cohesion between group members and a separation or
distancing from members outside the group. We have already noted that if we try and partition a complete graph into two factions than the density cost function will give a maximum fit for any partition. The reason is that we have thrown away the maximal concept of the clique definition and so we have not included everyone that could be included. An alternative cost function would be to expect no ties between the groups, we therefore have as the cost function the density of the number of connections between the groups. This cost function suffers from the same problems as the normal density function only on the absence rather than the presence of ties. The network with no edges has a perfect fit for any partition using this cost function. Ideally we would like to have the best of both measures. One approach would be to sum the measures taking care to make sure that both are either positive or negative. An alternative, but equivalent, approach is to measure the similarity of the partitioned data matrix to an 'ideal' structure matrix, this is the approach we shall explore.
By way of example let us consider a partition of a network into three groups. Applying the intuitive notions discussed in section 4.1 the idealised structure consists of three groups each of which is a complete component. We can use this information to construct the adjacency matrix of the idealised structure. Suppose that our three groups consist of 3,5 and 4 actors respectively and for clarity we will assume that actors 1,2 and 3 are in the first group; actors 4,5,6,7 and 8 are in the second group and actors 9,10,11 and 12 are in the last group. If this were not the case the we would simply re-arrange the rows and columns of the adjacency matrix so that the first three entries are in group 1 the second five in group 2 and so on. The resultant matrix has the following structure
011000000000101000000000110000000000000011110000000101110000000110110000000111010000000111100000000000000111000000001011000000001101000000001110
We can now compare this matrix with our data matrix to provide a measure of how good our partition is. There are a number of possible measures which compare matrices but we mention just two. Each will give rise to a cost function. The first is to simply the count the number of cells that are different between the data matrix and the idealised structure matrix the second method is to calculate the correlation coefficient between the matrices. (To do this we simply treat the entries of each matrix as if it is one long vector split over a number of rows). It must be remembered that since we do not have statistical independence then the correlation coefficient can have no statistical significance associated with it. Hence we are unable to interpret any particular value of the coefficient in absolute terms, we are, however, able to establish that one particular partition is closer to the ideal than another. Obviously every time there is a new distribution of actors between the groups then we need to construct a different idealised matrix and use this as the basis of our cost function.
We shall now take as an example the Karate Club data analysed in the previous section. We recall that our clique analysis assigned some of the actors into one of two possible groups but there was not sufficient separation of the other actors to place them into a particular group. We already know from the study that every actor joined one of the two clubs and so the data should split into two factions. The software package UCINET was run using the correlation cost function. The process was repeated a number of times using different starting configurations but the same groups always emerged. The following output was obtained
Group Assignments:
1: 1 2 3 4 5 6 7 8 10 11 12 13 14 17 18 20 22
2: 9 15 16 19 21 23 24 25 26 27 28 29 30 31 32 33 34
Grouped Adjacency Matrix
1 1 1 1 1 1 2 1 2 1 1 2 1 2 2 2 2 2 2 2 3 3 3 3 3
1 2 3 4 5 6 7 8 7 0 1 2 3 4 0 8 2 5 9 9 1 6 3 4 5 6 7 8 9 0 1 2 3 4
-----------------------------------------------------------------------
1 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 | 1 1 |
2 | 1 1 1 1 1 1 1 1 1 | 1 |
3 | 1 1 1 1 1 1 1 | 1 1 1 1 |
4 | 1 1 1 1 1 1 1 | |
5 | 1 1 1 1 | |
6 | 1 1 1 1 1 | |
7 | 1 1 1 1 1 | |
8 | 1 1 1 1 1 | |
17 | 1 1 1 | |
10 | 1 1 | 1 |
11 | 1 1 1 1 | |
12 | 1 1 | |
13 | 1 1 1 | |
14 | 1 1 1 1 1 | 1 |
20 | 1 1 1 | 1 |
18 | 1 1 1 | |
22 | 1 1 1 | |
-------------------------------------------------------------------------
15 | | 1 1 1 |
9 | 1 1 | 1 1 1 1 |
19 | | 1 1 1 |
21 | | 1 1 1 |
16 | | 1 1 1 |
23 | | 1 1 1 |
24 | | 1 1 1 1 1 1 |
25 | | 1 1 1 1 |
26 | | 1 1 1 1 |
27 | | 1 1 1 |
28 | 1 | 1 1 1 1 |
29 | 1 | 1 1 1 |
30 | | 1 1 1 1 1 |
31 | 1 | 1 1 1 1 |
32 | 1 | 1 1 1 1 1 1 |
33 | 1 | 1 1 1 1 1 1 1 1 1 1 1 1 |
34 | 1 1 1 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |
------------------------------------------------------------------------
The group assignments are in agreement with those reported by Zachary, both in terms of the factions identified by observation and subsequent club membership. Like Zachary we have placed 9 in the wrong group according to club membership after the split but this can be explained in terms of other overriding considerations not shared by other members (see Zachary 1977 for details).
4.6 Computational Considerations
All the techniques in this chapter require a computer to carry out any analysis. The algorithms employed all search the data (in different and often sophisticated ways) to find a solution to the problem. The time taken for any of these searches can increase exponentially with the size of the problem and this can be a practical issue. For the clique and k-plex analysis the important factors are the number of edges and the number of cliques or k-plexes. Since in general larger networks tend to be less dense the number of edges is not a big issue and a clique analysis on networks containing hundreds and even thousands of actors is feasible provided there are not too many groups. If there is a problem in terms of computation then the analyst should consider increasing the minimum size of a group (this significantly decreases the number of groups and sophisticated software can use this information to reduce the number of edges). For very large networks it is advisable to break them down into smaller portions before undertaking an analysis.
The combinatorial search routines have computation times that depend upon the number of actors and the number of factions. They are best suited to fewer than 100 actors and partitions into 5 or fewer groups. It would be computationally impossible to take say 1000 actors and partition them into 15 groups. Again the basic approach in dealing with large networks or many groups is to break the network down into smaller portions.
4.7 Performing a Cohesive Subgraph Analysis
In this section we give a step by step guide to examine the cohesive subgraph structure within a network. These steps are intended as a guide only and should not be seen as the definitive approach to analysing data. The guide is nave since we make no assumptions about the nature of the data and have no knowledge of the research questions being investigated. A sophisticated analysts can take the steps in a different order and/or by pass some of them.
STEP 1
If the data is binary go to step 2. If the data is valued then either use a technique for valued data, for example MDS or hierarchical clustering (provided it is symmetric) see chapter 2, or dichotomise. When dichotomising remember to make sure that a value of 1 should represent ties of a given strength together with ties of a stronger affiliation. This means taking all values greater than the prescribed value and recording them as 1 for similarity data but taking values less than the prescribed value for distance data. For multiple relations consider each relation separately. Once the data is in binary form go to step 2. Always repeat the analysis with different cut-off values to test for robustness and to get a more complete picture of the data.
STEP 2
Establish the components. For directed data find both the weak and strong components, see chapter 2. Components represent the simplest form of cohesive subgroup analysis and can sometimes provide sufficient information to answer research questions, if this is the case the analysis is completed. If the components do not provide sufficient information then for symmetric data go to step 3, directed data need to be symmetrized. Symmetrize by taking reciprocated ties, if the subsequent analysis fails then repeat by taking the underlying graph, if the underlying graph is used be careful in interpreting any cohesive subgroups that are found. It is good practice to find the strong components contained with any cohesive subgroup of an underlying graph, this means that within the cohesive subgroup everyone is at least mutually reachable. This can be achieved by simply taking the results of the component analysis and intersecting them with the results of the clique or k-plex analysis. Proceed to step 3.
STEP 3
If a partition is required and the number of groups are known or approximately known then go to step 6. Otherwise find all the cliques. If no or very few cliques are found then try the following:
if the minimum clique size is 4 or more then reduce it (but do not go below 3),
if the data was symmetrised and the reciprocal option was taken then construct the underlying graph,
if the data was dichotomised then reduce the cut-off value for similarity data ,or increase it for distance data.
If all these fail proceed to step 5 . If too many cliques are found (this may only be apparent after step 4) then try the reverse of the options above for too few cliques. First if you do not have reciprocated ties change this, then increase the minimum size (this will nearly always work) and finally change the cut-off value. If a simple listing of the cliques is sufficient then stop, but unless the structure is very simple it is worth going on to step 4.
STEP 4
Analyse the pattern of overlap. Note if there are a large number of cliques then the co-group method can be used to reduce them. Both methods should be used to try and identify major groupings of cliques and actors, outsiders and possible leaders of the groups. It should also be possible to deduce the approximate number of groupings and this information can be used to perform a factions analysis. If a partition of the data is required go to step 6 otherwise the analysis is complete.
STEP 5
Find all the k-plexes. Since this step is only being taken if all the clique steps have failed then the data should be in as weak a form as possible. If this is not the case and there are no or very few k-plexes then proceed as in step 3. If this fails increase k and repeat being careful to increase the minimum size requirement as specified in section 4.3. If this still fails then go to step 6. If a simple listing of k-plexes is sufficient then the analysis is complete, but like cliques analysing the pattern of overlap is recommended. In this case proceed to step 4 using the k-plexes as cliques.
STEP 6
Partition the network into factions. Factions should be implemented on each component of the network since separate components will always fall in different factions. Therefore only components which need to be broken into smaller groups need to be analysed. The analysts needs to decide how many factions should be used. This number can sometimes be deduced from the clique or k-plex analysis or is known from external sources. If there is no source of information then the analyst can try different numbers starting from 2 moving up to 3, 4 and 5. If the outcome of the analysis is one large group with a all other groups consisting of single or few actors then this indicates that the data cannot be split into this many groups. If repeated runs produce the same actors in the smaller groups then the routine is identifying outsiders. Remove the outsiders and repeat the analysis, outsiders can also be identified using the clique and k-plex methods. If this and all the other methods fail then it is reasonable to assume the data does not posses any cohesive subgroups.
4.8 An Example
We shall now give a full example of the steps outlined in the previous section on some data collected by Newcomb (1961) and Nordlie (1958) on preference rankings in a fraternity. Newcomb performed an experiment that consisted of creating a fraternity composed of seventeen mutually unacquainted undergraduate transfer students at the University of Michigan. in the fall of 1956. In return for board and lodgings each student supplied data over a sixteen week period, including a complete rank ordering of the other sixteen by 'favourableness of feeling'. We shall examine the data for week 15; rankings are from 1, most favourable, to 16, least favourable, no ties were allowed. The data is given below
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- --
1 0 12 15 5 10 11 6 4 7 16 8 9 2 3 13 14 1
2 8 0 13 2 3 6 9 10 5 15 7 4 11 12 14 16 1
3 8 11 0 10 12 3 5 13 4 14 6 2 9 15 7 16 1
4 6 4 15 0 3 2 10 11 5 16 9 8 7 14 12 13 1
5 5 4 13 2 0 8 10 6 1 14 12 11 3 9 15 16 7
6 6 9 14 3 8 0 7 1 2 15 13 11 4 10 12 16 5
7 12 4 8 6 14 10 0 5 9 16 2 1 7 11 13 15 3
8 1 9 15 3 6 4 13 0 11 14 10 8 2 7 12 16 5
9 10 5 13 3 7 1 12 9 0 16 11 6 8 4 14 15 2
10 2 12 14 11 10 6 3 4 7 0 9 1 15 13 5 16 8
11 9 3 6 4 7 13 5 14 8 16 0 2 10 11 12 15 1
12 8 2 12 7 11 14 1 10 3 16 5 0 6 9 15 13 4
13 1 10 14 9 8 5 3 2 7 15 12 11 0 6 13 16 4
14 4 9 16 10 15 2 8 11 1 14 3 7 6 0 12 13 5
15 12 8 11 3 16 7 9 13 4 14 15 5 6 10 0 2 1
16 12 5 16 3 11 8 7 15 2 14 9 1 13 10 6 0 4
17 4 3 14 2 6 10 9 11 1 16 8 7 5 13 12 15 0
STEP 1
Since the data is not binary we need to transform it before proceeding with an analysis (we shall deliberately not use the methods discussed in chapter 2). Rank ordering is problematic in terms of comparisons between actors, to see this we need only consider two extremes. If one actor basically likes everyone and another dislikes everyone but they are both constrained to place everyone else in a ranked order then the values will have widely different meanings. Since the data is historical we have little choice but to assume that there is some consistency, but if we were designing the study from scratch we would be well advised not to use rankings, at least not in this form. We shall start by assuming that the actor's top 5 choices are important and we therefore replace rankings from 1 to 5 with the value 1 and change all other values to zero. We shall change these values later to check that we have robust groupings. The data now looks like:
1 1 1 1 1 1 1 1
1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7
- - - - - - - - - - - - - - - - -
1 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0 0 1
2 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 0 1
3 0 0 0 0 0 1 1 0 1 0 0 1 0 0 0 0 1
4 0 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1
5 1 1 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0
6 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 1
7 0 1 0 0 0 0 0 1 0 0 1 1 0 0 0 0 1
8 1 0 0 1 0 1 0 0 0 0 0 0 1 0 0 0 1
9 0 1 0 1 0 1 0 0 0 0 0 0 0 1 0 0 1
10 1 0 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0
11 0 1 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1
12 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1
13 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1
14 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 1
15 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 1 1
16 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1
17 1 1 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0
STEP 2
Perform a component analysis. The data has only one weak component consisting of all the actors. There is one large strong component and four trivial strong components consisting of the singleton actors 3,10,15 and 16. This analysis has not provided us with any insight into the internal structure of the group. We therefore need to symmetrize the data and continue with a more detailed analysis. We symmetrize by taking reciprocated ties to obtain the matrix:
1 1 1 1 1 1 1 1
1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7
- - - - - - - - - - - - - - - - -
1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 1
2 0 0 0 1 1 0 0 0 1 0 0 1 0 0 0 0 1
3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
4 0 1 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1
5 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0
6 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0
7 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0
8 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0
9 0 1 0 1 0 1 0 0 0 0 0 0 0 1 0 0 1
10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
11 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0
12 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0
13 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1
14 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
17 1 1 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0
STEP 3
A standard clique analysis reveals 7 cliques namely:
1: 2 4 9 17
2: 2 4 5
3: 1 8 13
4: 1 13 17
5: 4 6 9
6: 6 8 13
7: 7 11 12
We note that actors 3,10,14,15 and 16 are not in any cliques. In fact a look at the symmetrised adjacency matrix reveals that these actors are isolates. This number of cliques clearly provides us with some information and we therefore proceed to step 4.
STEP 4
The group co-membership method yields a large group consisting of actors {1,2,4,5,6,8,9,13,17} with a smaller group of {7,11,12} and the outsiders. The co-group clustering agrees with this but provides some insight into the structure of the larger group, the larger group can be split into {2,4,5,6,9,17} and {1,6,8,13,17} indicating that actors 6 and 17 are important within this network.
A repeat analysis taking just the top 3 choices gives similar results. In this instance the groups reported come from the component analysis and a clique analysis is not required. Taking the top 7 choices produces similar but the outsiders tend to cloud the standard clique analysis. The Faction method with 3 groups however gives the groupings {1,2,4,5,6,8,9,13,14,17}, {10,15,16} and {3,7,11,12}, which agree very closely with our previous analysis. In this instance the {10,15,16} group had no ties with the other two groups. We therefore conclude that our analysis is robust and represents the structure inherent in the data and we terminate our analysis.
PAGE 6
EVERETT
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