Tyll Krueger

Degree distributions of evolving alphabetic bipartite networks and their projections

Many real-world systems are modeled as evolving bipartite networks and their one-mode projections. In particular, Discrete Combinatorial Systems (DCSs), which consist of a finite set of elementary units and different combinations of these units, can be modeled by a subclass of bipartite networks known as Alphabetic Bipartite Networks or @a-BiNs, where the bottom partite-set contains a fixed number of nodes (the elementary units) and the top partite-set grows unboundedly with time through the addition of nodes (the combinations). The principal questions in the study of @a-BiN evolution are to predict the degree distribution of the bottom set and that of the projection onto the bottom set (the bottom projection), from a knowledge of the bipartite growth dynamics. In this paper, we propose a realistic growth model for @a-BiNs, where the degree distribution of the top set (i.e., the distribution of the number of elementary units in the composite entities in the DCS) can be any arbitrary distribution with finite first and second moments. Utilizing an exact correspondence between the preferential growth of @a-BiNs and the Polya Urn scheme, we analytically solve the model to compute exact degree distributions of the bottom (fixed) set and the bottom projection. To the best of our knowledge, this is the first work which proposes and solves such a generalized growth model for @a-BiNs. We also derive that the degree distributions of both the bottom set and the bottom projection, suitably normalized with time, converge to distributions that are invariant over time. We also verify that this improved model can accurately explain the degree distributions of several real-world DCSs and their projections.

Understanding Evolution of Inter-Group Relationships Using Bipartite Networks

In online social systems, users with common affiliations or interests form social groups for discussing various topical issues. We study the relationships among these social groups, which manifest through users who are common members of multiple groups, and the evolution of these relationships as new users join the groups. Focusing on a certain number of the most popular groups, we model the group memberships of users as a subclass of bipartite networks, known as Alphabetic Bipartite Networks (α-BiNs), where one of the partitions contains a fixed number of nodes (the popular groups) while the other grows unboundedly with time (new users joining the groups). Specifically, we consider the evolution of the thresholded projection of the user-group bipartite network onto the set of groups, which accurately represents the inter-group relationships. We propose and solve a preferential attachment based growth model for evolution of α-BiNs, and analytically compute the degree distribution of the thresholded projection. We further investigate whether the predictions of this model can explain the projection degree distributions of user-group networks derived from several real social systems (Livejournal, Youtube and Flickr). The study also shows that the inter-group network is tightly knit, and there is an implicit semantic hierarchy within its structure, that is clearly identified by the method of thresholding. To the best of our knowledge, this is the first attempt to analytically model the dynamical relationships among groups in online social systems.

Optimizing topology in Bit Torrent based networks

In this paper, we discuss the importance of the network connectivities of the peers in Bit Torrent based systems in determining the download performance of the peers. In this context, assuming that the fraction of the peers of each bandwidth are known, we derive optimal connectivities of the peers that help to improve the average latency of the peers. We represent the topology of a Bit Torrent based system as a weighted graph, where the average edge weight of the graph directly relates to the download latency of the peers. We formulate the average edge weight of the whole system as a linear function of the fraction of the edges that connect peers of different bandwidth and derive the topology that maximizes the average edge weight of the network. Simulation results based on the Bit Torrent protocol validates the fact that in the optimal topology, peers have 13% better download latency as compared to topologies formed in the normal Bit Torrent based systems. Further the obtained topology also improves the fairness of the system as compared to normal Bit Torrent significantly.

Asymptotic shape and the speed of propagation of continuous-time continuous-space birth processes

Abstract We formulate and prove a shape theorem for a continuous-time continuous-space stochastic growth model under certain general conditions. Similar to the classical lattice growth models, the proof makes use of the subadditive ergodic theorem. A precise expression for the speed of propagation is given in the case of a truncated free-branching birth rate.

Threshold-based epidemic dynamics in systems with memory

In this article we analyze an epidemic dynamics model (SI) where we assume that there are k susceptible states, that is a node would require multiple

On the broadcast of segmented messages in dynamic networks

(k)

Chained Typical Subspaces - a Quantum Version of Breiman's Theorem

contacts before it gets infected. In specific, we provide a theoretical framework for studying diffusion rate in complete graphs and d -regular trees with extensions to dense random graphs. We observe that irrespective of the topology, the diffusion process could be divided into two distinct phases: i) the initial phase, where the diffusion process is slow, followed by ii) the residual phase where the diffusion rate increases manifold. In fact, the initial phase acts as an indicator for the total diffusion time in dense graphs. The most remarkable lesson from this investigation is that such a diffusion process could be controlled and even contained if acted upon within its initial phase.

Intergroup networks as random threshold graphs.

This paper makes a systematic attempt to understand the effect of message size on the speed and efficiency of message broadcast. It considers a realistic situation where a single message may be too large to be sent in over a single connection and hence might require to be transmitted in segments. In specific, we look into the push and pull message transfer techniques and investigate in details their effect on broadcast time as well as total number of redundant contacts incurred during the transmission of segmented messages. For such segmentation and a complete graph topology with n nodes, we observe that the time required for broadcast scales as nk-1/k (assuming there are k packets in one message segment) as opposed to log n in the single message epidemic case (k = 1). In order to improve broadcast time and reduce the number of useless contacts we propose different variants of the push and pull message transfer techniques. In this regard we introduce the concept of giveup, which allows a node to terminate broadcast on sensing its neighborhood has received the message. We further study the effect of message segmentation on various types of topologies like d-regular graph, random graph etc. and observe that even for simple push technique, there exists an optimal d for which the dynamics becomes fast. We also simulate our results on real traces and finally provide some suggestions for network designers which we believe will help in faster message dissemination and lesser wastage, especially in case of dynamic networks.