Temporal Ordered Clustering in Dynamic Networks: Unsupervised and Semi-supervised Learning Algorithms

In temporal ordered clustering, given a single snapshot of a dynamic network in which nodes arrive at distinct time instants, we aim at partitioning its nodes into $K$ ordered clusters $\mathcal{C}_1 \prec \cdots \prec \mathcal{C}_K$ such that for $i<j$, nodes in cluster $\mathcal{C}_i$ arrived before nodes in cluster $\mathcal{C}_j$, with $K$ being a data-driven parameter and not known upfront. Such a problem is of considerable significance in many applications ranging from tracking the expansion of fake news to mapping the spread of information. We first formulate our problem for a general dynamic graph, and propose an integer programming framework that finds the optimal clustering, represented as a strict partial order set, achieving the best precision (i.e., fraction of successfully ordered node pairs) for a fixed density (i.e., fraction of comparable node pairs). We then develop a sequential importance procedure and design unsupervised and semi-supervised algorithms to find temporal ordered clusters that efficiently approximate the optimal solution. To illustrate the techniques, we apply our methods to the vertex copying (duplication-divergence) model which exhibits some edge-case challenges in inferring the clusters as compared to other network models. Finally, we validate the performance of the proposed algorithms on synthetic and real-world networks.Comment: 14 pages, 9 figures, and 3 tables. This version is submitted to a journal. A shorter version of this work is published in the proceedings of IEEE International Symposium on Information Theory (ISIT), 2020. The first two authors contributed equall

Spatiotemporal dynamics on small-world neuronal networks: The roles of two types of time-delayed coupling

We investigate temporal coherence and spatial synchronization on small-world networks consisting of noisy Terman-Wang (TW) excitable neurons in dependence on two types of time-delayed coupling: $\{x_j(t-\tau)-x_i (t)\}$ and $\{x_j(t-\tau)-x_i(t-\tau)\}$. For the former case, we show that time delay in the coupling can dramatically enhance temporal coherence and spatial synchrony of the noise-induced spike trains. In addition, if the delay time $\tau$ is tuned to nearly match the intrinsic spike period of the neuronal network, the system dynamics reaches a most ordered state, which is both periodic in time and nearly synchronized in space, demonstrating an interesting resonance phenomenon with delay. For the latter case, however, we can not achieve a similar spatiotemporal ordered state, but the neuronal dynamics exhibits interesting synchronization transition with time delay from zigzag fronts of excitations to dynamic clustering anti-phase synchronization (APS), and further to clustered chimera states which have spatially distributed anti-phase coherence separated by incoherence. Furthermore, we also show how these findings are influenced by the change of the noise intensity and the rewiring probability. Finally, qualitative analysis is given to illustrate the numerical results.Comment: 17 pages, 9 figure

Relevance of Dynamic Clustering to Biological Networks

Network of nonlinear dynamical elements often show clustering of synchronization by chaotic instability. Relevance of the clustering to ecological, immune, neural, and cellular networks is discussed, with the emphasis of partially ordered states with chaotic itinerancy. First, clustering with bit structures in a hypercubic lattice is studied. Spontaneous formation and destruction of relevant bits are found, which give self-organizing, and chaotic genetic algorithms. When spontaneous changes of effective couplings are introduced, chaotic itinerancy of clusterings is widely seen through a feedback mechanism, which supports dynamic stability allowing for complexity and diversity, known as homeochaos. Second, synaptic dynamics of couplings is studied in relation with neural dynamics. The clustering structure is formed with a balance between external inputs and internal dynamics. Last, an extension allowing for the growth of the number of elements is given, in connection with cell differentiation. Effective time sharing system of resources is formed in partially ordered states.Comment: submitted to Physica D, no figures include

Understanding and modeling the small-world phenomenon in dynamic networks

The small-world phenomenon first introduced in the context of static graphs consists of graphs with high clustering coefficient and low shortest path length. This is an intrinsic property of many real complex static networks. Recent research has shown that this structure is also observable in dynamic networks but how it emerges remains an open problem. In this paper, we propose a model capable of capturing the small-world behavior observed in various real traces. We then study information diffusion in such small-world networks. Analytical and simulation results with epidemic model show that the small-world structure increases dramatically the information spreading speed in dynamic networks

A Fast and Efficient Incremental Approach toward Dynamic Community Detection

Community detection is a discovery tool used by network scientists to analyze the structure of real-world networks. It seeks to identify natural divisions that may exist in the input networks that partition the vertices into coherent modules (or communities). While this problem space is rich with efficient algorithms and software, most of this literature caters to the static use-case where the underlying network does not change. However, many emerging real-world use-cases give rise to a need to incorporate dynamic graphs as inputs. In this paper, we present a fast and efficient incremental approach toward dynamic community detection. The key contribution is a generic technique called $\Delta-screening$, which examines the most recent batch of changes made to an input graph and selects a subset of vertices to reevaluate for potential community (re)assignment. This technique can be incorporated into any of the community detection methods that use modularity as its objective function for clustering. For demonstration purposes, we incorporated the technique into two well-known community detection tools. Our experiments demonstrate that our new incremental approach is able to generate performance speedups without compromising on the output quality (despite its heuristic nature). For instance, on a real-world network with 63M temporal edges (over 12 time steps), our approach was able to complete in 1056 seconds, yielding a 3x speedup over a baseline implementation. In addition to demonstrating the performance benefits, we also show how to use our approach to delineate appropriate intervals of temporal resolutions at which to analyze an input network

Coupled Maps with Growth and Death: An Approach to Cell Differentiation

An extension of coupled maps is given which allows for the growth of the number of elements, and is inspired by the cell differentiation problem. The growth of elements is made possible first by clustering the phases, and then by differentiating roles. The former leads to the time sharing of resources, while the latter leads to the separation of roles for the growth. The mechanism of the differentiation of elements is studied. An extension to a model with several internal phase variables is given, which shows differentiation of internal states. The relevance of interacting dynamics with internal states (``intra-inter" dynamics) to biological problems is discussed with an emphasis on heterogeneity by clustering, macroscopic robustness by partial synchronization and recursivity with the selection of initial conditions and digitalization.Comment: LatexText,figures are not included. submitted to PhysicaD (1995,revised 1996 May