Brief Announcements / Back Matter

Motivated by the structure of social networks, this paper initiates a study of distributed algorithms in networks that exhibit a core-periphery structure. Such networks contain two distinct groups of nodes: a large and sparse, group identified as the periphery, which is loosely organized around a small, and densely connected group identified as the core. We identify four basic properties that are relevant to the interplay between core and periphery. For each of these properties, we propose a corresponding axiom that captures the behavior expected of a social network based on a core-periphery structure. We then address their usefulness for distributed computation, by considering a nontrivial algorithmic task of significance in both the distributed systems world and the social networks world, namely, the distributed construction of a minimum-weight spanning tree. 1. Axiomatic Approach. Most existing approaches to modeling social networks are based on creating a set of mathematical rules for gradually constructing a member of a given class of social networks (e.g., the Preferential Attachment model). A complementary approach that we promote, and adopt in this paper, is to base our characterization of social (and complex) networks in general, and their main structural components in particular, on axiomatic grounds, where the axioms are chosen so as to capture fundamental structural and computational properties of the modeled network. 2. Notations and Axioms. Let G(V,E) denote our network, where V is the set of nodes, |V | = n, and E is the set of edges, |E| = m. We consider a synchronous model of communication with messages of bounded (logarithmic) size, such as CONGEST model ([2]). Let N(v) denote the set of neighbors of node v ∈ V . For a set S ⊂ V and a node v ∈ S, let Nin(v, S) = N(v) ∩ S denote its set of neighbors within S and denote the number of neighbors of v in the set S by din(v, S) = |Nin(v, S)|. Analogously, let Nout(v, S) = N(v) ∩ V \ S denote v’s set of neighbors outside of S and let dout(v, S) = |Nout(v, S)|. For a set S ⊂ V , let ∂(S) be the edge boundary of S, namely the set of edges with one endpoint in S and the other outside of S and |∂(S)| = ∑ v∈S |Nout(v, S)|. We define the four properties of a core-periphery network. α-Influence. A subset of nodes S is α-Influential if |∂(S)| ≥ α ·m. Michael Borokhovich and Zvi Lotker were supported in part by the Israel Science Foundation (grant 894/09). David Peleg was supported in part by the Israel Science Foundation (grant 894/09), the United States-Israel Binational Science Foundation (grant 2008348), the I-CORE program of the Israel PBC and ISF (grant 4/11), and the Citi Foundation. Y. Afek (Ed.): DISC 2013, LNCS 8205, pp. 553–554, 2013. c © Springer-Verlag Berlin Heidelberg 2013