While questions about social cohesion lie at the core of our discipline, definitions are often vague and difficult to operationalize. We link research on social cohesion and social embeddedness by developing a conception of structural cohesion based on network nodeconnectivity. Structural cohesion is defined as the minimum number of actors who, if removed from a group, would disconnect the group. A structural dimension of embeddedness can then be defined through the hierarchical nesting of these cohesive structures. We demonstrate the empirical applicability of our conception of nestedness in two dramatically different substantive settings and discuss additional theoretical implications with reference to a wide array of substantive fields. "...social solidarity is a wholly moral phenomenon which by itself is not amenable to exact observation and especially not to measurement." (Durkheim, (1893 [1984], p.24) "The social structure [of the dyad] rests immediately on the one and on the other of the two, and the secession of either would destroy the whole. ... As soon, however, as there is a sociation of three, a group continues to exist even in case one of the members drops out." (Simmel (1908 [1950], p. 123)

Topology control problems are concerned with the assignment of power values to the nodes of an ad~hoc network so that the power assignment leads to a graph topology satisfying some specified properties. This paper considers such problems under several optimization objectives, including minimizing the maximum power and minimizing the total power. A general approach leading to a polynomial algorithm is presented for minimizing maximum power for a class of graph properties called monotone properties. The difficulty of generalizing the approach to properties that are not monotone is discussed. Problems involving the minimization of total power are known to be NP-complete even for simple graph properties. A general approach that leads to an approximation algorithm for minimizing the total power for some monotone properties is presented. Using this approach, a new approximation algorithm for the problem of minimizing the total power for obtaining a 2-node-connected graph is obtained. It is shown that this algorithm provides a constant performance guarantee. Experimental results from an implementation of the approximation algorithm are also presented.

We use random sampling as a tool for solving undirected graph problems. We show that the sparse graph, or skeleton, that arises when we randomly sample a graph’s edges will accurately approximate the value of all cuts in the original graph with high probability. This makes sampling effective for problems involving cuts in graphs. We present fast randomized (Monte Carlo and Las Vegas) algorithms for approximating and exactly finding minimum cuts and maximum flows in unweighted, undirected graphs. Our cut-approximation algorithms extend unchanged to weighted graphs while our weighted-graph flow algorithms are somewhat slower. Our approach gives a general paradigm with potential applications to any packing problem. It has since been used in a near-linear time algorithm for finding minimum cuts, as well as faster cut and flow algorithms. Our sampling theorems also yield faster algorithms for several other cut-based problems, including approximating the best balanced cut of a graph, finding a k-connected orientation of a 2k-connected graph, and finding integral multicommodity flows in graphs with a great deal of excess capacity. Our methods also improve the efficiency of some parallel cut and flow algorithms. Our methods also apply to the network design problem, where we wish to build a network satisfying certain connectivity requirements between vertices. We can purchase edges of various costs and wish to satisfy the requirements at minimum total cost. Since our sampling theorems apply even when the sampling probabilities are different for different edges, we can apply randomized rounding to solve network design problems. This gives approximation algorithms that guarantee much better approximations than previous algorithms whenever the minimum connectivity requirement is large. As a particular example, we improve the best approximation bound for the minimum k-connected subgraph problem from 1.85 to 1 � O(�log n)/k).

We consider the problem of deploying or repairing a sensor network to guarantee a specified level of multi-path connectivity (k-connectivity) between all nodes. Such a guarantee simultaneously provides fault tolerance against node failures and high overall network capacity (by the max-flow min-cut theorem). We design and analyze the first algorithms that place an almostminimum number of additional sensors to augment an existing network into a k-connected network, for any desired parameter k. Our algorithms have provable guarantees on the quality of the solution. Specifically, we prove that the number of additional sensors is within a constant factor of the absolute minimum, for any fixed k. We have implemented greedy and distributed versions of this algorithm, and demonstrate in simulation that they produce high-quality placements for the additional sensors.

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A wireless sensor network consists of many low-cost, low-power sensor nodes, which can perform sensing, simple computation, and transmission of sensed information. Long distance transmission by sensor nodes is not energy efficient, since energy consumption is a superlinear function of the transmission distance. One approach to prolong network lifetime while preserving network connectivity is to deploy a small number of costly, but more powerful, relay nodes whose main task is communication with other sensor or relay nodes. In this paper, we assume that sensor nodes have communication range r> 0 while relay nodes have communication range R ≥ r, and study two versions of relay node placement problems. In the first version, we want to deploy the minimum number of relay nodes so that between each pair of sensor nodes, there is a connecting path consisting of relay and/or sensor nodes. In the second version, we want to deploy the minimum number of relay nodes so that between each pair of sensor nodes, there is a connecting path consisting solely of relay nodes. We present a polynomial time 7-approximation algorithm for the first problem, and a polynomial time (5 + ɛ)-approximation algorithm for the second problem, where ɛ> 0 can be any given constant.

We present two new algorithms for the problem of nding a minimum-cost k-vertex connected spanning subgraph. The rst algorithm works on undirected graphs with at least 6k vertices and achieves an approximation of 6 times the kth harmonic number (which is O(log k)), The second algorithm works on any graph (directed or undirected) and gives an O( n=)-approximation algorithm for any > 0 and k (1 )n. These algorithms improve on the previous best approximation factor (more than k=2). The latter algorithm also extends to other problems in network design with vertex connectivity requirements. Our main tools are setpair relaxations, a theorem of Mader's (in the undirected case) and iterative rounding (general case).

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While questions about social cohesion lie at the core of our discipline, no clear definition of cohesion exists. We present a definition of social cohesion based on network connectivity that leads to an operationalization of social embeddedness. We define cohesiveness as the minimum number of actors who, if removed from a group, would disconnect the group. This definition generates hierarchically nested groups, where highly cohesive groups are embedded within less cohesive groups. We discuss the theoretical implications of this definition and demonstrate the empirical applicability of our conception of nestedness by testing the predicted correlates of our cohesion measure within high school friendship and interlocking directorate networks. Keywords: Social networks, social theory, social cohesion, connectivity algorithm, embeddedness. "...social solidarity is a wholly moral phenomenon which by itself is not amenable to exact observation and especially not to measurement." (Durkheim, (1893 [1984], p.24) "The social structure [of the dyad] rests immediately on the one and on the other of the two, and the secession of either would destroy the whole. ... As soon, however, as there is a sociation of three, a group continues to exist even in case one of the members drops out." (Simmel (1908 [1950], p. 123)

We present an approximation algorithm for solving graph problems in which a low-cost set of edges must be selected that has certain vertex-connectivity properties. In the survivable network design problem, one is given a value r ij for each pair of vertices i and j, and must find a minimum-cost set of edges such that there are r ij vertex-disjoint paths between vertices i and j. In the case for which r ij 2 f0; 1; 2g for all i; j, we can find a solution of cost no more than 3 times the optimal cost in polynomial time. In the case in which r ij = k for all i; j, we can find a solution of cost no more than 2H(k) times optimal, where H(n) = 1 + 1 2 + \Delta \Delta \Delta + 1 n . No approximation algorithms were previously known for these problems. Our algorithms rely on a primal-dual approach which has recently led to approximation algorithms for many edge-connectivity problems. 1 Introduction Let G = (V; E) be an undirected graph with non-negative costs c e 0 on all edges e 2 E. In...