The problem of increasing edge or vertex-connectivity of a given graph up to a specified target value k by adding the smallest number of new edges is called connectivity augmentation. These problems were first studied in 1976 by Eswaran and Tarjan [64] and Plesnik [275] and were shown to be polynomially solvable for k = 2. The problems have important applications such as the network construction problem [279], the rigidity problem in grid frameworks [13, 99], the data security problem [110, 172], and the rectangular dual graph problem in floor planning [303]. We refer to [81, 241] surveys for this study.

In this chapter, we mainly treat the edge-connectivity augmentation problem for a given target value k. For a general k, Watanabe and Nakamura [308] established in 1987 a min-max theorem, based on which they gave an O(k2(kn + m)n4) time algorithm. Afterward, Frank [78] gave a unified approach to various edgeconnectivity augmentation problems by making use of the edge-splitting theorems of Lovász [200, 202] and Mader [206, 208]. Then Nagamochi and Ibaraki [236] proposed an O((nm + n2 log n) log n) time algorithm by combining the minimumcut algorithm in Section 3.2 and the approach of Frank. If the graph under consideration is weighted by real numbers, this algorithm can be further simplified and can be extended to solve the edge-connectivity augmentation problem for the entire range of target k in O(nm + n2 log n) time [238], as will be explained in Section 8.4. By using extreme vertex sets in Section 1.5.3, Benczúr and Karger [20] gave an O(n2 log5n) time randomized algorithm of Monte Carlo type to optimally increase a multigraph.

The concept of extreme vertex sets, defined in Section 1.5.3, was first introduced by Watanabe and Nakamura [308] to solve the edge-connectivity augmentation problem. The fastest deterministic algorithm currently known for computing all extreme vertex sets was given by Naor, Gusfield, and Martel [259]. Their algorithm first computes the Gomory–Hu cut tree of the graph and then finds all maximal k-edge-connected components for some k, from which all extreme vertex sets are identified by Lemma 1.42, taking O(n(mn log(n2/m)) running time. Bencz&úr and Karger [20] have given a Monte Carlo–type randomized algorithm, which runs in O(n2 log5n) time but is rather involved. Notice that computing all extreme vertex sets is not easier than finding a minimum cut since at least one of the extreme vertex sets is a minimum cut.

In this chapter, we give a simple and efficient algorithm for computing all extreme vertex sets in a given graph, and we show some applications of extreme vertex sets. The algorithm will be used to solve the edge-connectivity augmentation problem in Section 8.3. In Section 6.1,we design a deterministic O(nm + n2 log n) time algorithm for computing the family χ(G) of extreme vertex sets in a given edge-weighted graph G, which is a laminar family, as observed in Lemma 1.41. As a new application of extreme vertex sets, in Section 6.2 we consider a dynamic graph G in which the weight of edges incident to a designated vertex may increase or decrease with time and we give a dynamic minimum cut algorithm that reports a minimum cut of the current G whenever the weight of an edge is updated.

In this chapter, we investigate structures and algorithms of cactus representations, which were introduced in Section 1.5.4 to represent all minimum cuts in an edgeweighted graph G. Throughout this chapter, we assume that λ(G) > 0 for a given graph G, which implies that G is connected. Let C(G) denote the set of all minimum cuts in G. In Section 5.1, we define a canonical form of cactus representations. In Section 5.2, we show that a subset of C(G) that consists of minimum cuts separating two given vertices, s and t, can be represented by a simple cactus structure. In Section 5.3, we design an O(mn + n2 log n) time algorithm for constructing a cactus representation R of C(G).

Canonical Forms of Cactus Representations

In this section, we discuss cactus representations for a subset of minimum cuts, and we prove the existence of two canonical forms, which we call the cycle-type and junction-type normal cactus representations. Such a canonical representation is useful in designing an efficient algorithm that constructs a cactus representation for all the minimum cuts of a given graph [244]. It also helps to efficiently test whether two given graphs have the same “structure” with respect to their minimum cuts, which is based on a planar isomorphism algorithm due to Hopcroft and Tarjan [126].

A cactus representation for a given subset C ⊆ C(G), if one exists, may not be unique unless we impose further structural restrictions.

In Chapter 3, we predicted and found that the perceptions in people's minds concerning whether a target individual was a friend of a prominent person significantly affected the target individual's reputation concerning work performance in an organization (Kilduff and Krackhardt, 1994). The actual existence of friendship links, recognized by both parties in each link, had no significant effect on other people's perceptions of an individual's reputation as a high performer. This research showed that people's perceptions of relations helped to determine reputations, whereas the actual structure of relations had no effect.

In this chapter, we focus again on perceptions of the friendship network, this time investigating how perceptions are shaped by preexisting expectations. We chose the friendship network to study because this network affects important choices individuals make. We ask, under what circumstances are individuals' perceptions of the friendship network shaped by schemas concerning how people typically behave in the friendship role?

The role of friend is well understood in society, as indicated by the high level of agreement within societies concerning how friends should act in relation to each other (Argyle and Henderson, 1985: 92). People have access to a schema or strategy that specifies how individuals typically act in this role (see the discussions in DiMaggio, 1991; Swidler, 1986). Cognitive psychologists have described schemas as mental structures that enable people to anticipate the general features of recurring situations (Neisser, 1976: 51–78).

In the previous two chapters, we showed that perceptions of social networks matter and that such perceptions are systematically biased. But some people are more accurate than others in perceiving network patterns. If this is so, do these accurate people gain benefits in organizational arenas of competition and power? This is the theme we investigate in this chapter. We expand the discussion to include perceptions of both friendship and advice networks, and investigate whether an accurate perception of the political landscape – including who are the central players — predicts who has power in the organization.

How does one assess the political landscape in an organization? One way of addressing this question is to identify the key political actors in the organization (Pfeffer, 1981). But simply identifying the most powerful actors may not give sufficient information to anticipate the overall dynamics of resistance and support for political acts. Additional questions about these actors come to mind: Are these powerful actors organized such that they tend to act in unison? Do they represent different political constituencies? Precisely whom does each have influence over? Beyond knowing who is powerful, it is useful to know how the powerful and powerless are organized or structured (Bailey, 1969: 108).

One way to approach the answers to these deeper questions about the political landscape is to study access to and the control of information flow in the organization (Pettigrew, 1973).

Good administrators sometimes fail to understand social structure and fail to anticipate its consequences for organizational survival. This can leave organizations vulnerable to manipulation by skilled political entrepreneurs. In one example, the entire top management team of a manufacturing company learned from a network analysis that the bomb threats, shootings, and vandalism threatening the future of the company were instigated by partisans of a lower-ranking manager, who had systematically recruited family, friends, and neighbors into the company over a thirty-year period. In a district desperate for jobs, these partisans felt loyalty to the informal leader, who had provided them information that allowed them to be first in line for vacancies on Monday morning. The CEO, confronted with an analysis of the deep cleavages existing in the social structure of the organization resulting from the informal patterns of recruiting over decades, had this to say about those who had been hired: “ … they just seemed like waves of turtles coming over the hill; hired as they made it to our door” (Burt, 1992: 1).

This story illustrates the gap at the heart of our understanding of organizational behavior. It illustrates how important it is for managers and would-be leaders to accurately perceive the network relations that connect people, and to actively manage these network relations. This story also illustrates how informal leaders who may lack formal authority can emerge to frustrate organizational functioning through the manipulation of network structures and the exercise of social influence.

An organization can be considered a socio-emotional system in which energy must be continually expended in order to keep the system on course (cf. Katz and Kahn, 1966). Among the many threats to system functioning are negative emotions. The workplace is a site where people often experience negative emotions associated with stress, anxiety, tension, and emotional pain (Basch and Fisher, 2000). Two classic experiments (Latané and Arrowood, 1963; Schachter, Willerman, Hyman, and Festinger, 1961) established that workers involved in all but the most routinized tasks who are subject to negative emotions tend to suffer decrements in the quality and quantity of their production. Further, these negative emotions correlate with individuals' negative work-related attitudes (Weiss and Cropanzano, 1996) and health problems (Frost, 2003: 3), and can prove contagious (Hatfield, Cacioppo, and Rapson, 1994) with deleterious effects for other employees' levels of cooperation and performance (Barsade, 2002). We know that some people become central actors in dealing with the negative emotions of colleagues (Frost, 2003; Frost and Robinson, 1999), but there is still little understanding of who these unusual people are. In this chapter, we spotlight the emotion helping network and its central players from a personality interaction perspective.

In this book, we have emphasized the distinctiveness of the individual in the context of the structuring of organizational social networks. This relationship between the micro and macro has proved elusive for network research. Thus, we have renewed the call to “bring the individual back in” when conducting structural analysis (Kilduff and Krackhardt, 1994). Our objective includes helping the next generation of network researchers understand the benefits of simultaneously considering individuals and social structures.

In this last chapter, we anticipate future directions for the research program described in this book. In looking to the future, we try to adopt some of the advantages and overcome some of the limitations of existing approaches to social network research. The influential structural hole perspective and similar work focused on actor centrality have brought a welcome focus on the agency of central individuals, but have tended to deliberately neglect the cognitions and personalities of actors in favor of an assumption of rational pursuit of personal advantage (Burt, 1992). By contrast, the new surge of work focused on small worlds is welcome in bringing an emphasis on dynamics to the network field, but too often this work tends to treat actors as pawns subject to all-powerful system forces (e.g., Dorogovtsev and Mendes, 2003). In looking to the future, we first review possible extensions of cognitive social network research and then explore topics related to the dynamic interplay of distinctive individuals in complex social networks in organizations.

The basic idea we investigate in this chapter is whether the relative importance, reputation, and value of any particular individual in an organization (in the eyes of others) are affected by the company the individual is perceived to keep. The assessment of reputation, we suggest, is likely to be enhanced if the individual is perceived to have a high-status friend. Becoming the friend of a high-status person is not easy, and those who are fortunate to enjoy such access are likely to gain considerable social capital. High-status people are carefully scrutinized to see who their associates are. This is not a new insight. Shakespeare's Falstaff, an intimate acquaintance of Crown Prince Harry in the Henry IV plays, is depicted as reveling in reflected glory. Certainly, the Baron de Rothschild (according to the anecdote in Chapter 1) was in no doubt concerning the value his apparent friendship would confer in terms of tangible financial capital becoming available from those impressed with his public social endorsement of the person with whom he walked “arm-in-arm.”

How Perceptions Affect Reputation

The theoretical framework within which we investigated the determinants of reputation in organizational labor markets was balance theory (Heider, 1958). From this perspective, someone perceived to be the friend of a positively valued other is also likely to be perceived positively: There is a strain toward cognitive balance in the perceptions of observers.

In the previous two chapters, we showed the relevance of social networks for the understanding of turnover and crises in organizations. But what enables organizations to promote coordination and collectivity? How do people with diverse backgrounds, goals, and values successfully coordinate their activities in organizations? The usual answer to these questions is that organizational culture provides the glue that keeps the organization together. But the organizational culture literature has neglected the importance of social connections in producing shared systems of meaning. In this chapter, we begin the process of remedying this oversight through an emphasis on how people who are tied to each other create locally shared cognitive understandings. First, we provide an in-depth analysis of how the diversity of cultural interpretations within one organization is controlled through the friendship network. Second, we extend the discussion to include how network embeddedness affects agreement concerning the structuring of networks across three different organizations.

Previous organizational culture research has tended to treat the culture of an organization as an independent variable that can be manipulated to control deviant behavior (e.g., Ouchi, 1980). From this culture-as-amanagerial-tool perspective, an effective organization is like a clan, in that it relies on mechanical solidarity – a religious adherence to common beliefs and practices – to ensure cooperation (Durkheim, 1933: 175–8). The clan cannot tolerate any divergence from the “totality of belief and sentiments common to all members of the group” (Durkheim, 1933: 129).

One of the enduring questions we face as human beings concerns why some people outcompete others in the race for life's prizes. In work organizations, for example, why are some people better performers than others? One answer to this question is provided by research on the importance of structural position. Within each specific work context, some individuals occupy more advantageous positions in social networks than other individuals. These positions allow access to people who are otherwise disconnected from each other. The individuals who act as go-betweens, bridging the “structural holes” between disconnected others, facilitate resource flows and knowledge sharing across the organization. Their contributions to organizational functioning may lead to enhanced rewards, including faster promotions (Burt, 1992) and higher performance ratings.

Research on structural position has emphasized the importance of being in the right place (Brass, 1984) but has neglected both the possibility that the network positions occupied by individuals might be influenced by their psychology and the possibility that personality and social network position might combine to influence important outcomes such as work performance. The structural approach to organizational dynamics tends to emphasize the structure of positions in social space (Blau, 1993; Pfeffer, 1991) and avoids dependence on difficult-to-measure psychological properties of actors (e.g., McPherson et al., 1992).