Vertex-pursuit in random directed acyclic graphs

We examine a dynamic model for the disruption of information flow in hierarchical social networks by considering the vertex-pursuit game Seepage played in directed acyclic graphs (DAGs). In Seepage, agents attempt to block the movement of an intruder who moves downward from the source node to a sink. The minimum number of such agents required to block the intruder is called the green number. We propose a generalized stochastic model for DAGs with given expected total degree sequence. Seepage and the green number is analyzed in stochastic DAGs in both the cases of a regular and power law degree sequence. For each such sequence, we give asymptotic bounds (and in certain instances, precise values) for the green number

Community tracking in a cMOOC and nomadic learner behavior identification on a connectivist rhizomatic learning network

This article contributes to the literature on connectivism, connectivist MOOCs (cMOOCs) and rhizomatic learning by examining participant interactions, community formation and nomadic learner behavior in a particular cMOOC, #rhizo15, facilitated for 6 weeks by Dave Cormier. It further focuses on what we can learn by observing Twitter interactions particularly. As an explanatory mixed research design, Social Network Analysis and content analysis were employed for the purposes of the research. SNA is used at the macro, meso and micro levels, and content analysis of one week of the MOOC was conducted using the Community of Inquiry framework. The macro level analysis demonstrates that communities in a rhizomatic connectivist networks have chaotic relationships with other communities in different dimensions (clarified by use of hashtags of concurrent, past and future events). A key finding at the meso level was that as #rhizo15 progressed and number of active participants decreased, interaction increased in overall network. The micro level analysis further reveals that, though completely online, the nature of open online ecosystems are very convenient to facilitate the formation of community. The content analysis of week 3 tweets demonstrated that cognitive presence was the most frequently observed, while teaching presence (teaching behaviors of both facilitator and participants) was the lowest. This research recognizes the limitations of looking only at Twitter when #rhizo15 conversations occurred over multiple platforms frequented by overlapping but not identical groups of people. However, it provides a valuable partial perspective at the macro meso and micro levels that contribute to our understanding of community-building in cMOOCs

Second-Order Agents on Ring Digraphs

The paper addresses the problem of consensus seeking among second-order linear agents interconnected in a specific ring topology. Unlike the existing results in the field dealing with one-directional digraphs arising in various cyclic pursuit algorithms or two-directional graphs, we focus on the case where some arcs in a two-directional ring graph are dropped in a regular fashion. The derived condition for achieving consensus turns out to be independent of the number of agents in a network.Comment: 6 pages, 10 figure

Nested hierarchies in planar graphs

We construct a partial order relation which acts on the set of 3-cliques of a maximal planar graph G and defines a unique hierarchy. We demonstrate that G is the union of a set of special subgraphs, named `bubbles', that are themselves maximal planar graphs. The graph G is retrieved by connecting these bubbles in a tree structure where neighboring bubbles are joined together by a 3-clique. Bubbles naturally provide the subdivision of G into communities and the tree structure defines the hierarchical relations between these communities

Multilayer Network of Language: a Unified Framework for Structural Analysis of Linguistic Subsystems

Recently, the focus of complex networks research has shifted from the analysis of isolated properties of a system toward a more realistic modeling of multiple phenomena - multilayer networks. Motivated by the prosperity of multilayer approach in social, transport or trade systems, we propose the introduction of multilayer networks for language. The multilayer network of language is a unified framework for modeling linguistic subsystems and their structural properties enabling the exploration of their mutual interactions. Various aspects of natural language systems can be represented as complex networks, whose vertices depict linguistic units, while links model their relations. The multilayer network of language is defined by three aspects: the network construction principle, the linguistic subsystem and the language of interest. More precisely, we construct a word-level (syntax, co-occurrence and its shuffled counterpart) and a subword level (syllables and graphemes) network layers, from five variations of original text (in the modeled language). The obtained results suggest that there are substantial differences between the networks structures of different language subsystems, which are hidden during the exploration of an isolated layer. The word-level layers share structural properties regardless of the language (e.g. Croatian or English), while the syllabic subword level expresses more language dependent structural properties. The preserved weighted overlap quantifies the similarity of word-level layers in weighted and directed networks. Moreover, the analysis of motifs reveals a close topological structure of the syntactic and syllabic layers for both languages. The findings corroborate that the multilayer network framework is a powerful, consistent and systematic approach to model several linguistic subsystems simultaneously and hence to provide a more unified view on language

Kernel-based Inference of Functions over Graphs

The study of networks has witnessed an explosive growth over the past decades with several ground-breaking methods introduced. A particularly interesting -- and prevalent in several fields of study -- problem is that of inferring a function defined over the nodes of a network. This work presents a versatile kernel-based framework for tackling this inference problem that naturally subsumes and generalizes the reconstruction approaches put forth recently by the signal processing on graphs community. Both the static and the dynamic settings are considered along with effective modeling approaches for addressing real-world problems. The herein analytical discussion is complemented by a set of numerical examples, which showcase the effectiveness of the presented techniques, as well as their merits related to state-of-the-art methods.Comment: To be published as a chapter in `Adaptive Learning Methods for Nonlinear System Modeling', Elsevier Publishing, Eds. D. Comminiello and J.C. Principe (2018). This chapter surveys recent work on kernel-based inference of functions over graphs including arXiv:1612.03615 and arXiv:1605.07174 and arXiv:1711.0930