As noted in the opening chapter, our aim in designing this book was to have a sustained examination of the general topic of network clustering. A clustering is a general term that means a set of clusters. Clustering also refers to a process for establishing a clustering. There are several types of clusterings, including a partition (a set of clusters that do not overlap and cover the whole set of units), a hierarchy (usually represented by a dendrogram), a pyramid, a fuzzy clustering, a clustering with overlapping clusters, and a clustering with disjoint clusters not covering the whole set of units. We use the more general term, clustering, throughout our discussion of the contributions contained in the book's chapters. However, when contributing authors use the terms partition and partitions, we use these terms.

Another goal for us was to make sure we included multiple perspectives and approaches to the problem of network clustering. As the foregoing chapters show, this topic is a highly diverse realm, both technically and substantively. We have much to learn from each other on both the technical and substantive fronts, at least when freed from the restrictions imposed by academic departments, fields, sub-fields, and different specific approaches. In an ideal world, generating knowledge transcends such constraints.

Even though many of the contributing authors ended their contributions with open problems, we use this concluding chapter to make additional suggestions regarding potential future work. Our suggestions and speculations take two forms. One is to focus on issues raised in the individual chapters. The other deals with linking ideas considered in separate chapters of this volume.

14.1 Issues Raised within Chapters

We start with Chapter 2. Given that the network clustering literature has expanding at a rapid rate and will continue to do so, it makes sense to obtain the citation network for this literature for 2020. No doubt the identified network will be much larger. More importantly, it will include additional fields, new contributors, new perspectives, and new issues. All will merit further attention. We think the coupling of ideas from multiple fields will be particularly important.

The authors of this chapter identified a set of nine link islands as shown in Figure 2.7. They are an example of a partition not including all the units in the network. While they focused on four of them, the others could be examined further. They all have distinct structures, which raises the question of whether the details of network structure in citation networks have import for the generation of scientific knowledge and its transmission over time. More generally, this concern could be folded into the general issue of the impacts of network structure on network processes and the design of networks to achieve specific objectives. This is a general topic of considerable importance for the study of clustering all networks. While three of these link islands had little to do directly with the core clustering topics of this book, the general issue remains: How does the structure of a citation network inform our understanding of how knowledge is generated and transmitted?

The authors, both in this chapter and in their earlier work [3], suggest that the institutional structure of science has a very large impact on the generation of scientific knowledge and the generation of scientific citation networks. In this context, journal citation networks seem particularly important. Some are controlled by publishers such as Elsevier, Springer, and Wiley. But many are sponsored by professional associations defined for disciplines and specific scientific interest groups. Many of these associations are focused on promoting their scientific interests and those of their members. This includes the nature of publishing strategies and the promotion of their journals. Journal citation networks seem worthy of greater attention. There are two distinct but overlapping aspects to this. One features journal-to-journal networks in specific areas of scientific inquiry, as described in Chapter 2, with the second being the study of journal networks across all disciplines. See, for example, [12] which is part of a long term effort studying journal-to-journal networks. Bibliographic coupling, a relatively old idea introduced in 1963 [11], also merits further attention, especially with the other link islands identified by these authors. Also, it will be desirable to do more with fractional bibliographic coupling, especially with the Jaccard islands they identify.

In Chapter 3, the optimization approach to the clustering problem was advocated. Using an appropriate criterion function, we can express our clustering goals, including the reduction of complexity, understanding network structures, and modeling networks. Together, optimization and the sought goals help define the nature of a “good” clustering. Additional knowledge about the specific clustering problem can be expressed using the feasibility predicate which defines the set of feasible clusterings.

Also mentioned in this chapter was that most of the existing social network clustering approaches are essentially based on structural equivalence. In [8] a generalized blockmodeling based on different types of equivalences is described. Current procedures for generalized blockmodeling can be applied to networks with up to some hundreds of nodes. An important task is to develop efficient methods for generalized blockmodeling of large sparse networks.

Regarding Chapter 4, we were intrigued by the content of Figure 4.2 because it illustrated two community partitions highlighting different aspects of networks as reflected in these partitions. This seems particularly important. We have long thought that multiple partitions of the same network have value and that the notion of having, or even wanting, a single partition of a network as the “best” one makes little sense. This idea also was expressed in multiple chapters in this book, albeit in different substantive contexts. If anything, this suggests that having multiple partitions of the same network has considerable merit and examining them closely is important for understanding the interplay of network structures and the network processes generating them. Of course, this observation extends to multiple clusterings of the same network.

Given that the authors of this chapter noted the value of examining differences among the four perspectives regarding community detection, as they outlined them, examining these differences further is another important task. One of their four approaches has a consideration of a dynamical perspective which merits further attention. This is important, and we comment further on this in the next section because the issue of dynamics was raised in multiple chapters. Such ideas need to be considered in conjunction.

Chapter 5 presented a different approach to the partitioning networks that were considered therein, one that is very fast. To examine how the algorithm works, the author used planted community structures to explore the operation of label propagation algorithms. This is an important idea not only in its own right but in a far more general context. While it is abundantly clear that networks have diverse structures, most often obscured in the construction of simple networks indices such as modularity and centrality measures, examining the global partition structure of networks is of great importance. This suggests a need to examine a more extended set of planted structures. In turn, this raises the issue of generating a catalog of network structures with different global forms from which planted structures could be selected. Many discussions in this book examined related ideas regardless of whether these structures are planted, used as demonstration examples, or were designed to examine certain structures in the context of networks clustering.

The inclusion of node preferences in Chapter 5 is important also and forms a step towards the inclusion of node properties in the network clustering algorithms explicitly considered therein. This is a line of inquiry that blockmodeling folk need to consider given their preoccupation with clustering networks without being attentive to nodal attributes. Depending on the substantive concerns of analysts, the set of constraints provided in this chapter could be expanded along with increasing the number of node preference regimes that could be included.

Two further items in Chapter 5 have the potential for opening new avenues of inquiry. One is to use label propagation methods on a much wider set of empirical networks, a topic to which we return later. The other is the consideration of having partitions with overlapping clusters and groups. This issue, having importance for blockmodeling as well as community detection, was raised in other chapters in this volume.

But as noted in Chapter 1, and discussed extensively in Chapter 6, clustering, or even mere partitioning – and studying – valued networks is far from being a straightforward task. Yet it must be done. The authors of Chapter 6 claim that when pre-specified blockmodels are needed, generalized blockmodeling approaches are preferable. While we agree, we would argue that more work is needed to construct a wider range of pre-specified blockmodels. This includes going far beyond using only structural equivalence to include many other equivalence types. The generalized blockmodeling approach is designed to facilitate the creation of many different types of equivalences. As a concern for network clustering is present in so many fields, it is highly likely that the specifics of network structures in these fields will generate the construction of new equivalence types. In general, such constructions must be driven by substantive concerns which will vary by fields.

The authors of Chapter 6 consider a very small number of valued networks, albeit to a very useful effect as the resulting partitions that were established are interesting both in terms of the partitions produced and the substantive interpretations that followed. However, in a much broader context, the number of considered valued networks must be expanded greatly both by those using the procedures outlined in this chapter and in the employment of other methods. As the authors note, their extension of partitioning networks is a natural - and necessary - extension. Partitioning and, more generally, clustering many more valued networks can only expand our understanding of dealing with valued networks.

At face value, the content of Chapter 7 may strike some readers as having limited importance by being confined to relatively small networks. But consistent with the idea that research design matters is the notion that data quality matters. It matters greatly how data are obtained, especially for recording data accurately and not discarding useful information. The authors focused on actor non-response and provided methods for dealing with this problem. This is very useful for recovering a full network accurately, especially for delineating the global structure of such networks. This line of inquiry has been applied also to the study of network centrality indices [13].

Yet attention to both errors in the recording of ties and item non-response may have even greater importance. Detecting actor non-response is straightforward. Discerning the presence of the other two forms of measurement error is much more difficult. Doing this is a task of great importance.

It will be straightforward to conduct studies of these types of measurement errors with techniques similar to those used in Chapter 7. The value of such efforts would depend on our ability to detect such measurement errors. One of the data sets considered in Chapter 7 came from a study asking questions about seeking advice from others and providing advice from others in an organization. If the data are accurate, the transpose of one relation would correspond to the other. Discrepancies between the two networks provide clues regarding item-specific measurement error. It would be useful to have a collection of data sets with such “reversed” relations to detect differences as measurement error and get an estimate of the amount of inaccuracy in the reporting of such ties.

Having high quality data is of critical importance when studying the structure of networks and the processes generating these structures. This applies to all network data sets regardless of their sizes. While it may be tempting to think that measurement error is irrelevant for large networks, especially very large networks, we think this view would be mistaken. As was shown in Chapter 2, it is critical to “clean” the data, even for large networks. If the techniques described in Chapter 7 cannot be extended easily to much larger networks, then other ways of assessing the presence of measurement error and treating such errors must be developed. Error-prone data are not a good source for understanding network structure nor for understanding the processes generating them.

While on the topic of data quality, a more general statement can be made. An important base for developing data analytic methods is the availability of a collection of data sets specific for each selected problem. Information about data sets for network analytic problems can be obtained, for example, at the Colorado Index of Complex Networks (ICON) [1]. Unfortunately for some combinations of network “dimensions” (mode, weighted, node attributes, linked, temporal, spatial) the corresponding data sets are very scarce or non-existent. For example, for the linked networks discussed in Chapter 10, only a few interesting network collections are available. This needs to be expanded. Also, to study and develop methods for temporal network clustering [2] it is necessary to have some temporal networks with node attributes.

Chapter 8 is devoted to partitioning signed networks. By taking a formal approach, the authors laid the foundations for further work in this area, foundations we hope others can build upon. In this context, further developments regarding weak structural balance will be useful. Within the blockmodeling approach, the authors note that the criterion function used for partitioning signed networks contains a parameter, , allowing for differential weighting of two types of inconsistencies. One is the presence of negative ties in positive blocks and the other concerns having positive ties in negative blocks. The formulation was a natural extension with being used most often. But it created what can be called the alpha problem: How can values for be selected? While using made considerable sense, it can be view as an arbitrary choice, especially if the numbers of negative and positive ties differ greatly. Using different values for this parameter, most often, leads to different partitions of signed networks regardless of the number of clusters. The problem is simple to state: Is there a principled way of selecting values for ? Some attention has been given to this but without any clear resolution emerging thus far.

The authors couple the partitioning of signed networks to community detection approaches. In doing so, they expand the concept of modularity, defined initially for unsigned networks, to deal with the presence of signed ties. This is particularly useful, and we look forward to future efforts extending this line of analysis. Similarly, using spectral methods and considering the constant Potts model was useful. A book review article of two books, produced by physicists studying networks, introduced the idea of “The Invasion of the Physicists” [4]. Bonacich was very clear that there were some good ideas in this literature, despite the very colonial claim that the physicists had invented a totally new field, called network science, to which members of the social network field needed to be attentive. There were two reactions to this invasion. One was outrage, a very narrow parochial response. The other was to think that the “old” social network analytic field needed to pay attention. Chapter 8 expresses this attentiveness.

In the design of this volume, only one chapter was devoted to signed networks. But this type of network was mentioned in other chapters. We consider this further in the next section when we look at links between many ideas expressed in different chapters of this volume.

The authors of Chapter 9 are most explicit in affirming the idea of making connections between different parts of the overall network clustering literature. Their idea of extending the modularity concept, defined in the community detection literature, to two-mode networks has great appeal. We hope that their approach to two-mode data and using both projections from such data will end, once and for all, the debates about the loss of information when projections are made. The authors make a connection to signed two-mode networks with suggestions for future work which we hope will be heeded. They also consider the use of spectral methods and advocate the use of two-mode stochastic blockmodels. Clearly these ideas make explicit links to other chapters when the use of spectral methods was raised.

Chapter 9 presents multiple partitions of a classic two-mode data set. This is fully consistent with our idea of having multiple partitions of the same network that have legitimacy if they can be interpreted in substantive terms. Our hope is that the ideas expressed in this chapter can be extended to larger – even much larger – two-mode networks. We concur with their view that “the complexities of this type of data in terms of collecting, analyzing, and interpreting remain challenging and deeply fascinating.” They have provided practitioners with some very useful advice.

Chapter 10 presented ideas associated with blockmodeling linked networks conceived as collections of different one-mode networks coupled through two-mode networks. The strong message of this chapter, after multiple comparisons, is that the true-linked approach for analyzing linked networks has great promise. As only two empirical examples were considered, it would be useful to have other such networks analyzed in the same fashion. One of the open problems outlined in Chapter 10 concerns combining multiple criteria, expressed as criterion functions. The author links this idea to other work in the literature on multi-criteria partitioning, another observation fully consistent with the overall inclusive and integrative approach stressed in this volume. On the topic of multi-criteria partitioning of networks, some work has been done that we think will proved useful (see [9] and [5]). We note that having one-mode networks for different time points opens this approach to temporal dynamics, which we consider later in this chapter. Another link is made to stochastic blockmodeling, the topic of Chapter 11.

In our view, it does not matter if readers take a frequentist approach or Bayesian approach to analyzing data. Adherents of both perspectives will learn a great deal from considering the contents of Chapter 11. While the prose in this chapter comes perilously close to insisting that the Bayesian approach is the only viable approach, we do not think this is the author's intent. If so, the frequentists need to pay close attention to the contents of this chapter.

One of the key, and in our view, fundamental ideas expressed in this chapter is the idea of coupling generative mechanisms, as parts of general processes, to the coarse-grained modular structure, regardless of how fine-grained or coarse-grained are such depictions. We would extend this to a general statement about the coupling of network structures and the processes generating them. This idea is particularly relevant for many blockmodeling aficionados focused on depicting the macro structure of networks without considering the underlying network processes generating the identified network structures. In general terms, this is particularly important.

It is abundantly clear that the notion of a “model” is critical, for models can vary greatly, and that considering variations in models is important when thinking about network clustering. Without doubt, as the author of this chapter notes, having a multiplicity of models is a strength - but not only in a probabilistic sense. We cleave to a view that having multiple clustering models fitted to a network is important and that multiple such clusterings have the potential to suggest important ideas about network structure. Discerning the “best” such model may be a quixotic quest. But we agree with the author that any delineated clustering must have solid evidence that the fitted model is appropriate and justified.

Chapter 12, as noted in the opening chapter, picks up the idea of delineating coarse-grained structures of networks but within an explicit dynamic perspective. Their focus is on the “rich interplay between network structure and dynamics acting on top of the network” as a way of gauging the dynamical behavior of a network system. While we know that empirical networks have actors joining the network and other actors departing, it seems very useful to consider changes in the properties of actors located in a specific network. Not all problems can be solved at the same time. Examining change over a fixed network has considerable merit, especially if this can be generalized, as the authors note.

As a general framework for studying change, using a differential equation model for continuous data, or a difference equation model for discrete data, is especially appropriate. The authors of Chapter 12 focus on consensus dynamics in a specific empirical network along with a discussion of random walks in networks. Coupling them as dual processes is especially useful. It is well known that there are slow processes with a relaxed time scale for their operation and fast processes where the dynamics operate far more quickly. At face value, this applies across the entire network.

The idea of having a modular partitioned network for which the time scales for different parts of a network differ is especially intriguing. The authors consider a specific modular structure in the form of a diagonal blockmodel. This could be extended to other blockmodel structures. The ideas presented in Chapter 12 on the dynamics for signed networks are considered in more detail in the next section. The idea of incorporating dynamical processes to reveal network structure, as outlined in this chapter, has additional appeal. While the substantive content of Chapter 13 is focused on scientific coauthorship networks, the issue of dynamics is present also. The technical context is blockmodeling with a focus on identifying cores and discerning their stability over time as well as the instability of cores. The presence of cores is a critical feature of the structure of coauthorship relations within scientific communities. The presentation of useful indices for measuring the stability of cores is particularly useful, as are the visualizations of the movement of researchers between well-identified positions in a co-authorship network. It seems reasonable to think that these tools could be applied fruitfully to many other types of networks when there are temporal changes.

14.2 Linking Ideas Found in Different Chapters

A wide variety of approaches and methods related to network clustering have been presented in different chapters. While they could be viewed as rivals, it seems more fruitful to think of them as sources for ideas that could be coupled in a fruitful fashion. One general idea is to think of identifying, or creating, data sets where the different methods and algorithms could be used to cluster networks. Of course, there will be no one network for which all the methods could be applied. Differences regarding valued ties, signed ties, the number of relations studied, and the sizes of networks makes this impossible. But for various network types, a subset of the methods presented in this book could be mobilized. This notion can be coupled to the idea of having a catalog of different structures for which different methods, as shown in this volume, could be applied to useful effect.

The goal would not be to find a so-called winner but to examine the insights generated by each of the methods. As we know, every approach has its assumptions and some constraints implied by these assumptions. It follows that using as many different methods as possible allows us to assess the value of the clusterings that are obtained. They can be compared in a systematic fashion. A core feature of this effort must be grounded in the substantive concerns specific to the fields within which network data are collected. Establishing clusterings is not the final product, for they must be interpreted to help researchers understand the structures of networks and, if possible, the processes generating them, or outcomes generated over networks.

Perhaps a less ambitious path would be to combine ideas from different methods to construct different ways of thinking about tools and establishing additional methods. As a modest step in this direction, we consider linking ideas expressed in the specific chapters of this book. In our view, there is some value in drawing clear distinctions between different approaches to better understand both their strengths and limitations.

There may be researchers who think that any approach not labeled as community detection has little value. And there are others thinking that blockmodeling covers all possible network clustering approaches. Both views are mistaken. We have doubts about universalistic claims for single approaches. There are claims that community detection covers everything [10]. Some authors of this volume note that community detection is a special case of blockmodeling, a view on which we have commented already. Others claim that the whole field of network clustering is subsumed within stochastic blockmodeling [14].

Chapter 2 and 4 consider some of these issues. Clearly, blockmodeling and community detection have some common features. They also have different literatures, as documented in Chapter 2, suggesting there are some real differences between them. We doubt this is merely a matter of perception. It would be useful to have clear exploration of their similarities and differences to establish a succinct statement about them. The goal would not be to see which is “better”, a foolish quest, but to explore ways of combining ideas from both approaches to strengthen each of them.

One technique used in Chapter 2 was the identification of coherent parts of networks without establishing a partition of the entire network. Both the community detection and blockmodeling approaches seek partitions of entire networks. We think it would be useful to compare the results of identifying link islands and complete partitions to couple the interpretations that each approach yields. There may be other ways of identifying smaller parts of networks that could be included in such comparisons.

The authors of Chapters 4 and 9 examine closely the role of modularity as it pertains to network clustering. In doing so, they expand the formulation of this concept and provide additional operational equations. This idea is considered in other chapters also. It would be useful to develop these ideas further in a sustained effort. It seems important to examine whether modularity could be used in the formulation of some criterion functions used in blockmodeling. Considering whether some of the blockmodeling criterion functions could be used to reformulate definitions of modularity would be another avenue for exploration. Again, the objective is one of having ideas flow between different approaches.

Many chapters consider criterion functions that are optimized when delineating clusterings, with a wide variety of them being employed. It would be useful to examine the ways that variations across them have an impact on the clusters that are established. Also, the relationships among different criterion functions need to be studied in detail.

Chapter 5 examined label propagation with a suggestion that this approach could be used for signed networks. Chapter 8 focused on clustering signed networks. Could label propagation be useful for solving some of the clustering problems for signed networks? We certainly hope so. Chapter 8 settled for partitions consistent with strong balance, leaving open the issue of using weak balance for such networks in future work. One example in the literature [7] showed there were severe problems with the standard signed blockmodeling approach due to its handling of positive ties. The signed community detection approach had major problems with its handling of negative ties. Label propagation for signed networks has the potential to solve both problems.

There is a clear divide between deterministic approaches to network clustering and probabilistic methods. Chapter 11 is resolutely within the latter approach using Bayesian ideas. The authors of Chapter 9 and 12 make explicit links to stochastic blockmodeling, an idea meriting detailed attention. More generally, it would be useful to have a systematic assessment of the results stemming from the uses of deterministic and probabilistic approaches. Again, the goal would not be to establish which is the better approach but to see if there are complementary results and to explore any differences to see why the clustering results differ.

We turn to consider network dynamics. Chapter 11 and 12 are focused on dynamic change, albeit in very different ways. In thinking about change in networks, the mechanisms involved in network processes are critical. Being able to specify them and understand their operation has the utmost importance. This is the clear message in both chapters despite taking starkly different approaches. In the spirit of wanting to couple different ideas, we wonder if this can be done with these two approaches. At face value, this could be useful even though the technical issues involved will be fearsome. A step in this direction is provided in [6].

The authors of Chapter 12 tackle also signed networks within the rubric of structural balance. This is an ambitious approach with the potential of being very fruitful. However, it requires abandoning the notion that the network is fixed. The study of signed networks involves the examination of changes in the signs and strengths of ties. If actors drop out of such networks, or others join, the task becomes far more complex. We note that Chapter 4 also has a dynamic component that could be incorporated into this discussion.

14.3 A Brief Summary and Conclusion

Many ideas and issues have been raised throughout the book. As noted above, many avenues have been opened for future work on network clustering. We finish by stressing two very general ideas. One is the importance of the exchange of ideas between different approaches with the goal of strengthening them. The second is the coupling of network processes and network structures to help us understand both. Our hope is that this book will help promote these general issues as well as all the ideas contained in its chapters. If it helps in doing this, then this volume will have the impact we hoped it would have.

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