In many social networks, there is a high correlation between the similarity of actors and the existence of relationships between them. This paper introduces a model of network evolution where actors are assumed to have a small aversion from being connected to others who are dissimilar to themselves, and yet no actor strictly prefers a segregated network. This model is motivated by Schelling's [Schelling TC (1969) Models of segregation. Am Econ Rev 59:488-493] classic model of residential segregation, and we show that Schelling's results also apply to the structure of networks; namely, segregated networks always emerge regardless of the level of aversion. In addition, we prove analytically that attribute similarity among connected network actors always reaches a stationary distribution, and this distribution is independent of network topology and the level of aversion bias. This research provides a basis for more complex models of social interaction that are driven in part by the underlying attributes of network actors and helps advance our understanding of why dysfunctional social network structures may emerge. The Problem of Segregation in Social NetworksCooperation and conflict in social networks are central features of many contemporary social and policy issues, including commons governance, climate change policy, and economic development. Of particular importance is the well-known tendency for network linkages to concentrate between actors who are similar to one another in terms of certain key attributes-a phenomenon we refer to as "attribute closeness." Attribute closeness is one important indicator of segregation within a social network, as it signals tightly knit communities of homogenous actors and may reinforce divisions between disparate groups. These types of networks have been observed in a wide variety of contexts (1-3), including diverse examples such as race-and gender-oriented segregation in high school friendship networks and value-and belief-oriented segregation in environmental policy networks. Thus, the literature provides many empirical examples of segregation in terms of the structure of relationships, in addition to the geographically explicit residential segregation studied in Schelling's classic model of this phenomenon (4, 5).Network segregation is problematic when actors are faced with the need to collectively address complex social, environmental, or economic dilemmas. For example, social and policy networks are an important part of the machinery of collective action (6, 7). If the networks that form around social dilemmas are heavily fragmented, then it may be difficult for actors to develop the social capital necessary for the emergence of cooperative behavior (8). These fragmentations are observed in real-world policy networks, such as in regional planning networks where segregation is often observed between actors working in different functional domains (e.g., when transportation planners fail to coordinate with landuse planners), levels of government (e.g., when federal agencies ...
Abstract. A circuit cover of an edge-weighted graph (G, p) is a multiset of circuits in G such that every edge e is contained in exactly p(e) circuits in the multiset. A nonnegative integer valued weight vector p is admissible if the total weight of any edge-cut is even, and no edge has more than half the total weight of any edge-cut containing it. A graph G has the circuit cover property if (G, p) has a circuit cover for every admissible weight vector p . We prove that a graph has the circuit cover property if and only if it contains no subgraph homeomorphic to Petersen's graph. In particular, every 2-edge-connected graph with no subgraph homeomorphic to Petersen's graph has a cycle double cover.
Abstract:The odd edge connectivity of a graph G, denoted by o (G), is the size of a smallest odd edge cut of the graph. Let S be any given surface and be a positive real number. We proved that there is a function f S () (depends on the surface S and lim !0 f S () ¼ 1) such that any graph G embedded in S with the odd-edge connectivity at least f S () admits a nowhere-zero circular (2 þ )-flow. Another major result of the work is a new vertex splitting lemma which maintains the old edge connectivity and graph embedding. ß
The odd-girth of a graph is the length of a shortest odd circuit. A conjecture by Pavol Hell about circular coloring is solved in this article by showing that there is a function f ( ) for each : 0 < < 1 such that, if the odd-girth of a planar graph G is at least f ( ), then G is (2 + )-colorable. Note that the function f ( ) is independent of the graph G and → 0 if and only if f ( ) → ∞. A key lemma, called the folding lemma, is proved that provides a reduction method, which maintains the odd-girth of planar graphs. This lemma is expected to have applications in related problems.
The main result of the papzer is that any planar graph with odd girth at least 10k À 7 has a homomorphism to the Kneser graph G 2k1 k , i.e. each vertex can be colored with k colors from the set f1; 2;. .. ; 2k 1g so that adjacent vertices have no colors in common. Thus, for example, if the odd girth of a planar graph is at least 13, then the graph has a homomorphism to G 5 2 , also known as the Petersen graph. Other similar results for planar graphs are also obtained with better bounds and additional restrictions.
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