Probabilistic Inductive Classes of Graphs

Models of complex networks are generally defined as graph stochastic processes in which edges and vertices are added or deleted over time to simulate the evolution of networks. Here, we define a unifying framework - probabilistic inductive classes of graphs - for formalizing and studying evolution of complex networks. Our definition of probabilistic inductive class of graphs (PICG) extends the standard notion of inductive class of graphs (ICG) by imposing a probability space. A PICG is given by: (1) class B of initial graphs, the basis of PICG, (2) class R of generating rules, each with distinguished left element to which the rule is applied to obtain the right element, (3) probability distribution specifying how the initial graph is chosen from class B, (4) probability distribution specifying how the rules from class R are applied, and, finally, (5) probability distribution specifying how the left elements for every rule in class R are chosen. We point out that many of the existing models of growing networks can be cast as PICGs. We present how the well known model of growing networks - the preferential attachment model - can be studied as PICG. As an illustration we present results regarding the size, order, and degree sequence for PICG models of connected and 2-connected graphs.Comment: 15 pages, 6 figure

Dynamics of Social Balance on Networks

We study the evolution of social networks that contain both friendly and unfriendly pairwise links between individual nodes. The network is endowed with dynamics in which the sense of a link in an imbalanced triad--a triangular loop with 1 or 3 unfriendly links--is reversed to make the triad balanced. With this dynamics, an infinite network undergoes a dynamic phase transition from a steady state to "paradise"--all links are friendly--as the propensity p for friendly links in an update event passes through 1/2. A finite network always falls into a socially-balanced absorbing state where no imbalanced triads remain. If the additional constraint that the number of imbalanced triads in the network does not increase in an update is imposed, then the network quickly reaches a balanced final state.Comment: 10 pages, 7 figures, 2-column revtex4 forma

Visualizing Structural Balance in Signed Networks

Network visualization has established as a key complement to network analysis since the large variety of existing network layouts are able to graphically highlight different properties of networks. However, signed networks, i.e., networks whose edges are labeled as friendly (positive) or antagonistic (negative), are target of few of such layouts and none, to our knowledge, is able to show structural balance, i.e., the tendency of cycles towards including an even number of negative edges, which is a well-known theory for studying friction and polarization. In this work we present Structural-balance-viz: a novel visualization method showing whether a connected signed network is balanced or not and, in the latter case, how close the network is to be balanced. Structural-balance-viz exploits spectral computations of the signed Laplacian matrix to place network's nodes in a Cartesian coordinate system resembling a balance (a scale). Moreover, it uses edge coloring and bundling to distinguish positive and negative interactions. The proposed visualization method has characteristics desirable in a variety of network analysis tasks: Structural-balance-viz is able to provide indications of balance/polarization of the whole network and of each node, to identify two factions of nodes on the basis of their polarization, and to show their cumulative characteristics. Moreover, the layout is reproducible and easy to compare. Structural-balance-viz is validated over synthetic-generated networks and applied to a real-world dataset about political debates confirming that it is able to provide meaningful interpretations

The interplay of microscopic and mesoscopic structure in complex networks

Not all nodes in a network are created equal. Differences and similarities exist at both individual node and group levels. Disentangling single node from group properties is crucial for network modeling and structural inference. Based on unbiased generative probabilistic exponential random graph models and employing distributive message passing techniques, we present an efficient algorithm that allows one to separate the contributions of individual nodes and groups of nodes to the network structure. This leads to improved detection accuracy of latent class structure in real world data sets compared to models that focus on group structure alone. Furthermore, the inclusion of hitherto neglected group specific effects in models used to assess the statistical significance of small subgraph (motif) distributions in networks may be sufficient to explain most of the observed statistics. We show the predictive power of such generative models in forecasting putative gene-disease associations in the Online Mendelian Inheritance in Man (OMIM) database. The approach is suitable for both directed and undirected uni-partite as well as for bipartite networks

The Citation Field of Evolutionary Economics

Evolutionary economics has developed into an academic field of its own, institutionalized around, amongst others, the Journal of Evolutionary Economics (JEE). This paper analyzes the way and extent to which evolutionary economics has become an interdisciplinary journal, as its aim was: a journal that is indispensable in the exchange of expert knowledge on topics and using approaches that relate naturally with it. Analyzing citation data for the relevant academic field for the Journal of Evolutionary Economics, we use insights from scientometrics and social network analysis to find that, indeed, the JEE is a central player in this interdisciplinary field aiming mostly at understanding technological and regional dynamics. It does not, however, link firmly with the natural sciences (including biology) nor to management sciences, entrepreneurship, and organization studies. Another journal that could be perceived to have evolutionary acumen, the Journal of Economic Issues, does relate to heterodox economics journals and is relatively more involved in discussing issues of firm and industry organization. The JEE seems most keen to develop theoretical insights

Dynamics of MOOC Discussion Forums

In this integrated study of dynamics in MOOCs discussion forums, we analyze the interplay of temporal patterns, discussion content, and the social structure emerging from the communication using mixed methods. A special focus is on the yet under-explored aspect of time dynamics and influence of the course structure on forum participation. Our analyses show dependencies between the course structure (video opening time and assignment deadlines) and the overall forum activity whereas such a clear link could only be partially observed considering the discussion content. For analyzing the social dimension we apply role modeling techniques from social network analysis. While the types of user roles based on connection patterns are relatively stable over time, the high fluctuation of active contributors lead to frequent changes from active to passive roles during the course. However, while most users do not create many social connections they can play an important role in the content dimension triggering discussions on the course subject. Finally, we show that forum activity level can be predicted one week in advance based on the course structure, forum activity history and attributes of the communication network which enables identification of periods when increased tutor supports in the forum is necessary

Detecting groups of similar components in complex networks

We study how to detect groups in a complex network each of which consists of component nodes sharing a similar connection pattern. Based on the mixture models and the exploratory analysis set up by Newman and Leicht (Newman and Leicht 2007 {\it Proc. Natl. Acad. Sci. USA} {\bf 104} 9564), we develop an algorithm that is applicable to a network with any degree distribution. The partition of a network suggested by this algorithm also applies to its complementary network. In general, groups of similar components are not necessarily identical with the communities in a community network; thus partitioning a network into groups of similar components provides additional information of the network structure. The proposed algorithm can also be used for community detection when the groups and the communities overlap. By introducing a tunable parameter that controls the involved effects of the heterogeneity, we can also investigate conveniently how the group structure can be coupled with the heterogeneity characteristics. In particular, an interesting example shows a group partition can evolve into a community partition in some situations when the involved heterogeneity effects are tuned. The extension of this algorithm to weighted networks is discussed as well.Comment: 14 pages, 10 figures, latex, more discussions added, typos cleare