Community detection in directed acyclic graphs

Some temporal networks, most notably citation networks, are naturally represented as directed acyclic graphs (DAGs). To detect communities in DAGs, we propose a modularity for DAGs by defining an appropriate null model (i.e., randomized network) respecting the order of nodes. We implement a spectral method to approximately maximize the proposed modularity measure and test the method on citation networks and other DAGs. We find that the attained values of the modularity for DAGs are similar for partitions that we obtain by maximizing the proposed modularity (designed for DAGs), the modularity for undirected networks and that for general directed networks. In other words, if we neglect the order imposed on nodes (and the direction of links) in a given DAG and maximize the conventional modularity measure, the obtained partition is close to the optimal one in the sense of the modularity for DAGs.Comment: 2 figures, 7 table

Reduced-Dimension Linear Transform Coding of Correlated Signals in Networks

A model, called the linear transform network (LTN), is proposed to analyze the compression and estimation of correlated signals transmitted over directed acyclic graphs (DAGs). An LTN is a DAG network with multiple source and receiver nodes. Source nodes transmit subspace projections of random correlated signals by applying reduced-dimension linear transforms. The subspace projections are linearly processed by multiple relays and routed to intended receivers. Each receiver applies a linear estimator to approximate a subset of the sources with minimum mean squared error (MSE) distortion. The model is extended to include noisy networks with power constraints on transmitters. A key task is to compute all local compression matrices and linear estimators in the network to minimize end-to-end distortion. The non-convex problem is solved iteratively within an optimization framework using constrained quadratic programs (QPs). The proposed algorithm recovers as special cases the regular and distributed Karhunen-Loeve transforms (KLTs). Cut-set lower bounds on the distortion region of multi-source, multi-receiver networks are given for linear coding based on convex relaxations. Cut-set lower bounds are also given for any coding strategy based on information theory. The distortion region and compression-estimation tradeoffs are illustrated for different communication demands (e.g. multiple unicast), and graph structures.Comment: 33 pages, 7 figures, To appear in IEEE Transactions on Signal Processin

Making communities show respect for order

In this work we give a community detection algorithm in which the communities both respects the intrinsic order of a directed acyclic graph and also finds similar nodes. We take inspiration from classic similarity measures of bibliometrics, used to assess how similar two publications are, based on their relative citation patterns. We study the algorithm’s performance and antichain properties in artificial models and in real networks, such as citation graphs and food webs. We show how well this partitioning algorithm distinguishes and groups together nodes of the same origin (in a citation network, the origin is a topic or a research field). We make the comparison between our partitioning algorithm and standard hierarchical layering tools as well as community detection methods. We show that our algorithm produces different communities from standard layering algorithms

4Ward: a Relayering Strategy for Efficient Training of Arbitrarily Complex Directed Acyclic Graphs

Thanks to their ease of implementation, multilayer perceptrons (MLPs) have become ubiquitous in deep learning applications. The graph underlying an MLP is indeed multipartite, i.e. each layer of neurons only connects to neurons belonging to the adjacent layer. In contrast, in vivo brain connectomes at the level of individual synapses suggest that biological neuronal networks are characterized by scale-free degree distributions or exponentially truncated power law strength distributions, hinting at potentially novel avenues for the exploitation of evolution-derived neuronal networks. In this paper, we present ``4Ward'', a method and Python library capable of generating flexible and efficient neural networks (NNs) from arbitrarily complex directed acyclic graphs. 4Ward is inspired by layering algorithms drawn from the graph drawing discipline to implement efficient forward passes, and provides significant time gains in computational experiments with various Erd\H{o}s-R\'enyi graphs. 4Ward not only overcomes the sequential nature of the learning matrix method, by parallelizing the computation of activations, but also addresses the scalability issues encountered in the current state-of-the-art and provides the designer with freedom to customize weight initialization and activation functions. Our algorithm can be of aid for any investigator seeking to exploit complex topologies in a NN design framework at the microscale

Formal vs self-organised knowledge systems: a network approach

In this work we consider the topological analysis of symbolic formal systems in the framework of network theory. In particular we analyse the network extracted by Principia Mathematica of B. Russell and A.N. Whitehead, where the vertices are the statements and two statements are connected with a directed link if one statement is used to demonstrate the other one. We compare the obtained network with other directed acyclic graphs, such as a scientific citation network and a stochastic model. We also introduce a novel topological ordering for directed acyclic graphs and we discuss its properties in respect to the classical one. The main result is the observation that formal systems of knowledge topologically behave similarly to self-organised systems.Comment: research pape