Abstract. We present a new stochastic model for complex networks, based on a spatial embedding of the nodes, called the Spatial Preferred Attachment (SPA) model. In the SPA model, nodes have influence regions of varying size, and new nodes may only link to a node if they fall within its influence region. The spatial embedding of the nodes models the background knowledge or identity of the node, which will influence its link environment. In our model, nodes can determine their link environment based only on local knowledge of the network. We prove that our model gives a power law in-degree distribution, with exponent in [2, ∞) depending on the parameters, and with concentration for a wide range of in-degree values. We show that the model allows for edges that span a large distance in the underlying space, modelling a feature often observed in real-world complex networks.

Abstract. We present a new stochastic model for complex networks, based on a spatial embedding of the nodes, called the Spatial Preferred Attachment (SPA) model. In the SPA model, nodes have influence regions of varying size, and new nodes may only link to a node if they fall within its influence region. The spatial embedding of the nodes models the background knowledge or identity of the node, which will influence its link environment. In our model, nodes can determine their link environment based only on local knowledge of the network. We prove that our model gives a power law in-degree distribution, with exponent in [2, ∞) depending on the parameters, and with concentration for a wide range of in-degree values. We show that the model allows for edges that span a large distance in the underlying space, modelling a feature often observed in real-world complex networks.

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 ...

We present a deterministic model for online social networks (OSNs) based on transitivity and local knowledge in social interactions. In the iterated local transitivity (ILT) model, at each time step and for every existing node x, a new node appears that joins to the closed neighbor set of x. The ILT model provably satisfies a number of both local and global properties that have been observed in OSNs and other real-world complex networks, such as a densification power law, decreasing average distance, and higher clustering than in random graphs with the same average degree. Experimental studies of social networks demonstrate poor expansion properties as a consequence of the existence of communities with low numbers of intercommunity edges. Bounds on the spectral gap for both the adjacency and normalized Laplacian matrices are proved for graphs arising from the ILT model indicating such bad expansion properties. The cop and domination numbers are shown to remain the same as those of the graph from the initial time step G0, and the automorphism group of G0 is a subgroup of the automorphism group of graphs generated at all later time steps. A randomized version of the ILT model is presented that exhibits a tunable densification power-law exponent and maintains several properties of the deterministic model.

The size-Ramsey numberR(F ) of a graph F is the smallest integer m such that there exists a graph G on m edges with the property that any colouring of the edges of G with two colours yields a monochromatic copy of F . In this paper, first we focus on the size-Ramsey number of a path P n on n vertices. In particular, we show that 5n/2 − 15/2 ≤R(P n ) ≤ 74n for n sufficiently large. (The upper bound uses expansion properties of random d-regular graphs.) This improves the previous lower bound,R(P n ) ≥ (1 + √ 2)n − O(1), due to Bollobás, and the upper bound,R(P n ) ≤ 91n, due to Letzter. Next we study long monochromatic paths in edge-coloured random graph G(n, p) with pn → ∞. Let α > 0 be an arbitrarily small constant. Recently, Letzter showed that a.a.s. any 2-edge colouring of G(n, p) yields a monochromatic path of length (2/3 − α)n, which is optimal. Extending this result, we show that a.a.s. any 3-edge colouring of G(n, p) yields a monochromatic path of length (1/2 − α)n, which is also optimal. In general, we prove that for r ≥ 4 a.a.s. any r-edge colouring of G(n, p) yields a monochromatic path of length (1/r − α)n. We also consider a related problem and show that for any r ≥ 2, a.a.s. any r-edge colouring of G(n, p) yields a monochromatic connected subgraph on (1/(r−1)−α)n vertices, which is also tight.