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In this paper, we present a framework for convolutional coded Poisson receivers (CCPRs) that incorporates spatially coupled methods into the architecture of coded Poisson receivers (CPRs). We use density evolution equations to track the packet decoding process with the successive interference cancellation (SIC) technique. We derive outer bounds for the stability region of CPRs when the underlying channel can be modeled by a $\phi$-ALOHA receiver. The stability region is the set of loads that every packet can be successfully received with a probability of 1. Our outer bounds extend those of the spatially-coupled Irregular Repetition Slotted ALOHA (IRSA) protocol and apply to channel models with multiple traffic classes. For CCPRs with a single class of users, the stability region is reduced to an interval. Therefore, it can be characterized by a percolation threshold. We study the potential threshold by the potential function of the base CPR used for constructing a CCPR. In addition, we prove that the CCPR is stable under a technical condition for the window size. For the multiclass scenario, we recursively evaluate the density evolution equations to determine the boundaries of the stability region. Numerical results demonstrate that the stability region of CCPRs can be enlarged compared to that of CPRs by leveraging the spatially-coupled method. Moreover, the stability region of CCPRs is close to our outer bounds when the window size is large., Comment: Part of this work was presented in 2023 IEEE International Symposium on Information Theory (ISIT) [1] and 2024 IEEE International Symposium on Information Theory (ISIT) [2]

Recent advances in Large Language Models (LLMs) have demonstrated the emergence of capabilities (learned skills) when the number of system parameters and the size of training data surpass certain thresholds. The exact mechanisms behind such phenomena are not fully understood and remain a topic of active research. Inspired by the skill-text bipartite graph model proposed by Arora and Goyal for modeling semantic languages, we develop a mathematical theory to explain the emergence of learned skills, taking the learning (or training) process into account. Our approach models the learning process for skills in the skill-text bipartite graph as an iterative decoding process in Low-Density Parity Check (LDPC) codes and Irregular Repetition Slotted ALOHA (IRSA). Using density evolution analysis, we demonstrate the emergence of learned skills when the ratio of the number of training texts to the number of skills exceeds a certain threshold. Our analysis also yields a scaling law for testing errors relative to this ratio. Upon completion of the training, the association of learned skills can also be acquired to form a skill association graph. We use site percolation analysis to derive the conditions for the existence of a giant component in the skill association graph. Our analysis can also be extended to the setting with a hierarchy of skills, where a fine-tuned model is built upon a foundation model. It is also applicable to the setting with multiple classes of skills and texts. As an important application, we propose a method for semantic compression and discuss its connections to semantic communication., Comment: V1 was submitted to ISIT 2024 on Jan. 28, 2024. V2 was uploaded to ArXiv on April 13, 2024. V3 was uploaded to ArXiv on May 16, 2024

This research paper delves into the integration of OpenAI's ChatGPT into embodied agent systems, evaluating its influence on interactive decision-making benchmark. Drawing a parallel to the concept of people assuming roles according to their unique strengths, we introduce InterAct. In this approach, we feed ChatGPT with varied prompts, assigning it a numerous roles like a checker and a sorter, then integrating them with the original language model. Our research shows a remarkable success rate of 98% in AlfWorld, which consists of 6 different tasks in a simulated household environment, emphasizing the significance of proficient prompt engineering. The results highlight ChatGPT's competence in comprehending and performing intricate tasks effectively in real-world settings, thus paving the way for further advancements in task planning.

Various recent experimental results show that large language models (LLM) exhibit emergent abilities that are not present in small models. System performance is greatly improved after passing a certain critical threshold of scale. In this letter, we provide a simple explanation for such a phase transition phenomenon. For this, we model an LLM as a sequence-to-sequence random function. Instead of using instant generation at each step, we use a list decoder that keeps a list of candidate sequences at each step and defers the generation of the output sequence at the end. We show that there is a critical threshold such that the expected number of erroneous candidate sequences remains bounded when an LLM is below the threshold, and it grows exponentially when an LLM is above the threshold. Such a threshold is related to the basic reproduction number in a contagious disease., Comment: 5 pages, 1 figure

In this paper we study a susceptible infectious recovered (SIR) model with asymptomatic patients, contact tracing and isolation on a configuration network. Using degree based approximation, we derive a system of differential equations for this model. This system can not be solved analytically. We present an early-time analysis for the model. The early-time analysis produces an epidemic threshold. On one side of the threshold, the disease dies out quickly. On the other side, a significant fraction of the population are infected. The threshold only depends on the parameters of the disease, the mean access degree of the network, and the fraction of asymptomatic patients. The threshold does not depend on the parameter of contact tracing and isolation policy. We present an approximate analysis which greatly reduces computational complexity. The nonlinear system derived from the approximation is not almost linear. We present a stability analysis for this system. We simulate the SIR model with contact tracing and isolation on five real-world networks. Simulation results show that contact tracing and isolation are useful to contain epidemics.

Most existing works on analyzing the performance of a random ensemble of low-density parity-check (LDPC) codes assume that the degree distributions of the two ends of a randomly selected edge are independent. In the paper, we take one step further and consider ensembles of LDPC codes with degree-degree correlations. For this, we propose two methods to construct an ensemble of degree-degree correlated LDPC codes. We then derive a system of density evolution equations for such degree-degree correlated LDPC codes over a binary erasure channel (BEC). By conducting extensive numerical experiments, we show how the degree-degree correlation affects the performance of LDPC codes. Our numerical results show that LDPC codes with negative degree-degree correlation could improve the maximum tolerable erasure probability. Moreover, increasing the negative degree-degree correlation could lead to better unequal error protection (UEP) design., Comment: accepted by the 2023 IEEE International Symposium on Information Theory (ISIT)

The multichannel rendezvous problem is a fundamental problem for neighbor discovery in many IoT applications. The existing works in the literature focus mostly on improving the worst-case performance, and the average-case performance is often not as good as that of the random algorithm. As IoT devices (users) are close to each other, their available channel sets, though they might be different, are similar. Using the locality-sensitive hashing (LSH) technique in data mining, we propose channel hopping algorithms that exploit the similarity between the two available channel sets to increase the rendezvous probability. For the synchronous setting, our algorithms have the expected time-to-rendezvous (ETTR) inversely proportional to a well-known similarity measure called the Jaccard index. For the asynchronous setting, we use dimensionality reduction to speed up the rendezvous process. Our numerical results show that our algorithms can outperform the random algorithm in terms of ETTR.

In this paper we consider two resource allocation problems of URLLC traffic and eMBB traffic in uplink 5G networks. We propose to divide frequencies into a common region and a grant-based region. Frequencies in the grant-based region can only be used by eMBB traffic, while frequencies in the common region can be used by eMBB traffic as well as URLLC traffic. In the first resource allocation problem we propose a two-player game to address the size of the grant-based region and the size of the common region. We show that this game has specific pure Nash equilibria. In the second resource allocation problem we determine the number of packets that each eMBB user can transmit in a request-grant cycle. We propose a constrained optimization problem to minimize the variance of the number of packets granted to the eMBB users. We show that a water-filling algorithm solves this constrained optimization problem. From simulation, we show that our scheme, consisting of resource allocation according to Nash equilibria of a game, persistent random retransmission of URLLC packets and allocation of eMBB packets by a water-filling algorithm, works better than four other heuristic methods.

Recently, a novel coded compressed sensing (CCS) approach was proposed in [1] for dealing with the scalability problem for large sensing matrices in massive machine-type communications. The approach is to divide the compressed sensing (CS) problem into smaller CS sub-problems. However, such an approach requires stitching the results from the sub-problems to recover the result in the original CS problem. For this stitching problem, we propose a hierarchical stitching algorithm that is easier to implement in hardware for parallelization than the tree coding algorithm in [1]. For our algorithm, we also derive an upper bound on the probability of recovery errors., Comment: 5 pages, 3 figures

In this paper we present a generalized configuration model with random triadic closure (GCTC). This model possesses five fundamental properties: large clustering coefficient, power law degree distribution, short path length, non-zero Pearson degree correlation, and existence of community structures. We analytically derive the Pearson degree correlation coefficient and the clustering coefficient of the proposed model. We select a few datasets of real-world networks. By simulation, we show that the GCTC model matches very well with the datasets in terms of Pearson degree correlations and clustering coefficients. We also test three well-known community detection algorithms on our model, the datasets and other three prevalent benchmark models. We show that the GCTC model performs equally well as the other three benchmark models. Finally, we perform influence diffusion on the GCTC model using the independent cascade model and the linear threshold model. We show that the influence spreads of the GCTC model are much closer to those of the datasets than the other benchmark models. This suggests that the GCTC model is a suitable tool to study network science problems where degree correlation or clustering plays an important role.

We propose a periodic channel hopping (CH) sequence, called PPoL (Packing the Pencil of Lines in a finite projective plane), for the multichannel rendezvous problem. When $N-1$ is a prime power, its period is $N^2-N+1$, and the number of distinct rendezvous channels of PPoL is at least $N-2$ for any nonzero clock drift. By channel remapping, we construct CH sequences with the maximum time-to-rendezvous (MTTR) bounded by $N^2+3N+3$ if the number of commonly available channels is at least two. This achieves a roughly 50% reduction of the state-of-the-art MTTR bound in the literature., Comment: 5 pages

The group testing approach that achieves significant cost reduction over the individual testing approach has received a lot of interest lately for massive testing of COVID-19. Many studies simply assume samples mixed in a group are independent. However, this assumption may not be reasonable for a contagious disease like COVID-19. Specifically, people within a family tend to infect each other and thus are likely to be positively correlated. By exploiting positive correlation, we make the following two main contributions. One is to provide a rigorous proof that further cost reduction can be achieved by using the Dorfman two-stage method when samples within a group are positively correlated. The other is to propose a hierarchical agglomerative algorithm for pooled testing with a social graph, where an edge in the social graph connects frequent social contacts between two persons. Such an algorithm leads to notable cost reduction (roughly 20%-35%) compared to random pooling when the Dorfman two-stage algorithm is applied., Comment: 14 pages, 8 figures, submitted for publication

In comparison with individual testing, group testing (also known as pooled testing) is more efficient in reducing the number of tests and potentially leading to tremendous cost reduction. As indicated in the recent article posted on the US FDA website, the group testing approach for COVID-19 has received a lot of interest lately. There are two key elements in a group testing technique: (i) the pooling matrix that directs samples to be pooled into groups, and (ii) the decoding algorithm that uses the group test results to reconstruct the status of each sample. In this paper, we propose a new family of pooling matrices from packing the pencil of lines (PPoL) in a finite projective plane. We compare their performance with various pooling matrices proposed in the literature, including 2D-pooling, P-BEST, and Tapestry, using the two-stage definite defectives (DD) decoding algorithm. By conducting extensive simulations for a range of prevalence rates up to 5%, our numerical results show that there is no pooling matrix with the lowest relative cost in the whole range of the prevalence rates. To optimize the performance, one should choose the right pooling matrix, depending on the prevalence rate. The family of PPoL matrices can dynamically adjust their column weights according to the prevalence rates and could be a better alternative than using a fixed pooling matrix.

The network embedding problem that maps nodes in a graph to vectors in Euclidean space can be very useful for addressing several important tasks on a graph. Recently, graph neural networks (GNNs) have been proposed for solving such a problem. However, most embedding algorithms and GNNs are difficult to interpret and do not scale well to handle millions of nodes. In this paper, we tackle the problem from a new perspective based on the equivalence of three constrained optimization problems: the network embedding problem, the trace maximization problem of the modularity matrix in a sampled graph, and the matrix factorization problem of the modularity matrix in a sampled graph. The optimal solutions to these three problems are the dominant eigenvectors of the modularity matrix. We proposed two algorithms that belong to a special class of graph convolutional networks (GCNs) for solving these problems: (i) Clustering As Feature Embedding GCN (CAFE-GCN) and (ii) sphere-GCN. Both algorithms are stable trace maximization algorithms, and they yield good approximations of dominant eigenvectors. Moreover, there are linear-time implementations for sparse graphs. In addition to solving the network embedding problem, both proposed GCNs are capable of performing dimensionality reduction. Various experiments are conducted to evaluate our proposed GCNs and show that our proposed GCNs outperform almost all the baseline methods. Moreover, CAFE-GCN could be benefited from the labeled data and have tremendous improvements in various performance metrics., Comment: This manuscript was submitted to IEEE Transactions on Neural Networks and Learning Systems (IEEE TNNLS) on September 22, 2020, and is being under reviewed. This work was supported by the Ministry of Science and Technology (MOST) of Taiwan (R.O.C.) under Project MOST108-2221-E007-016-MY3. (Corresponding author: Ping-En Lu.)

Motivated by the need to hide the complexity of the physical layer from performance analysis in a layer 2 protocol, a class of abstract receivers, called Poisson receivers, was recently proposed in [1] as a probabilistic framework for providing differentiated services in uplink transmissions in 5G networks. In this paper, we further propose a deterministic framework of ALOHA receivers that can be incorporated into the probabilistic framework of Poisson receivers for analyzing coded multiple access with successive interference cancellation. An ALOHA receiver is characterized by a success function of the number of packets that can be successfully received. Inspired by the theory of network calculus, we derive various algebraic properties for several operations on success functions and use them to prove various closure properties of ALOHA receivers, including (i) ALOHA receivers in tandem, (ii) cooperative ALOHA receivers, (iii) ALOHA receivers with traffic multiplexing, and (iv) ALOHA receivers with packet coding. By conducting extensive simulations, we show that our theoretical results match extremely well with the simulation results.

In this paper, we develop a probabilistic framework for analyzing coded random access. Our framework is based on a new abstract receiver (decoder), called a Poisson receiver, that is characterized by a success probability function of a tagged packet subject to a Poisson offered load. We show that various coded slotted ALOHA (CSA) systems are Poisson receivers. Moreover, Poisson receivers have two elegant closure properties: (i) Poisson receivers with packet routing are still Poisson receivers, and (ii) Poisson receivers with packet coding are still Poisson receivers. These two closure properties enable us to use smaller Poisson receivers as building blocks for analyzing a larger Poisson receiver. As such, we can analyze complicated systems that are not possible by the classical tree evaluation method. In particular, for CSA systems with both spatial diversity and temporal diversity, we can use the framework of Poisson receivers to compute the exact (asymptotic) throughput. We demonstrate that our framework can be used to provide differentiated services between ultra-reliable low-latency communication (URLLC) traffic and enhanced mobile broadband (eMBB) traffic. By conducting extensive simulations, we also verify that our theoretical results match extremely well with the simulation results.

In this paper, we conduct mathematical and numerical analyses to address the following crucial questions for COVID-19: (Q1) Is it possible to contain COVID-19? (Q2) When will be the peak and the end of the epidemic? (Q3) How do the asymptomatic infections affect the spread of disease? (Q4) What is the ratio of the population that needs to be infected to achieve herd immunity? (Q5) How effective are the social distancing approaches? (Q6) What is the ratio of the population infected in the long run? For (Q1) and (Q2), we propose a time-dependent susceptible-infected-recovered (SIR) model that tracks 2 time series: (i) the transmission rate at time t and (ii) the recovering rate at time t. Such an approach is more adaptive than traditional static SIR models and more robust than direct estimation methods. Using the data provided by China, we show that the one-day prediction errors for the numbers of confirmed cases are almost in 3%, and the total number of confirmed cases is precisely predicted. Also, the turning point, defined as the day that the transmission rate is less than the recovering rate can be accurately predicted. After that day, the basic reproduction number $R_0$ is less than 1. For (Q3), we extend our SIR model by considering 2 types of infected persons: detectable and undetectable infected persons. Whether there is an outbreak in such a model is characterized by the spectral radius of a 2 by 2 matrix that is closely related to $R_0$. For (Q4), we show that herd immunity can be achieved after at least 1-1/$R_0$ fraction of individuals being infected. For (Q5) and (Q6), we analyze the independent cascade (IC) model for disease propagation in a configuration random graph. By relating the propagation probabilities in the IC model to the transmission rates and recovering rates in the SIR model, we show 2 approaches of social distancing that can lead to a reduction of $R_0$., Comment: 18 pages, 12 figures, 2 table, Parts of the codes used in this paper will be placed on GitHub (https://github.com/PingEnLu/Time-dependent_SIR_COVID-19), The latest version will be placed on http://gibbs1.ee.nthu.edu.tw/A_TIME_DEPENDENT_SIR_MODEL_FOR_COVID_19.PDF

In this paper we present a generalization of the classical configuration model. Like the classical configuration model, the generalized configuration model allows users to specify an arbitrary degree distribution. In our generalized configuration model, we partition the stubs in the configuration model into b blocks of equal sizes and choose a permutation function h for these blocks. In each block, we randomly designate a number proportional to q of stubs as type 1 stubs, where q is a parameter in the range [0; 1]. Other stubs are designated as type 2 stubs. To construct a network, randomly select an unconnected stub. Suppose that this stub is in block i. If it is a type 1 stub, connect this stub to a randomly selected unconnected type 1 stub in block h(i). If it is a type 2 stub, connect it to a randomly selected unconnected type 2 stub. We repeat this process until all stubs are connected. Under an assumption, we derive a closed form for the joint degree distribution of two random neighboring vertices in the constructed graph. Based on this joint degree distribution, we show that the Pearson degree correlation function is linear in q for any fixed b. By properly choosing h, we show that our construction algorithm can create assortative networks as well as disassortative networks. We present a percolation analysis of this model. We verify our results by extensive computer simulations., Comment: 23 pages, 4 figures

In the literature, there are several well-known periodic channel hopping (CH) sequences that can achieve maximum rendezvous diversity in a cognitive radio network (CRN). For a CRN with $N$ channels, it is known that the period of such a CH sequence is at least $N^2$. The asymptotic approximation ratio, defined as the ratio of the period of a CH sequence to the lower bound $N^2$ when $N \to \infty$, is still 2.5 for the best known CH sequence in the literature. An open question in the multichannel rendezvous problem is whether it is possible to construct a periodic CH sequence that has the asymptotic approximation ratio 1. In this paper, we tighten the theoretical gap by proposing CH sequences, called IDEAL-CH, that have the asymptotic approximation ratio 2. For a weaker requirement that only needs the two users to rendezvous on one commonly available channel in a period, we propose channel hopping sequences, called ORTHO-CH, with period $(2p +1)p$, where $p$ is the smallest prime not less than $N$.

In this paper, we consider the multichannel rendezvous problem in cognitive radio networks (CRNs) where the probability that two users hopping on the same channel have a successful rendezvous is a function of channel states. The channel states are modelled by two-state Markov chains that have a good state and a bad state. These channel states are not observable by the users. For such a multichannel rendezvous problem, we are interested in finding the optimal policy to minimize the expected time-to-rendezvous (ETTR) among the class of {\em dynamic blind rendezvous policies}, i.e., at the $t^{th}$ time slot each user selects channel $i$ independently with probability $p_i(t)$, $i=1,2, \ldots, N$. By formulating such a multichannel rendezvous problem as an adversarial bandit problem, we propose using a reinforcement learning approach to learn the channel selection probabilities $p_i(t)$, $i=1,2, \ldots, N$. Our experimental results show that the reinforcement learning approach is very effective and yields comparable ETTRs when comparing to various approximation policies in the literature., Comment: 5 pages, 9 figures. arXiv admin note: text overlap with arXiv:1906.10424

In this paper, we consider the multichannel rendezvous problem in cognitive radio networks (CRNs) where the probability that two users hopping on the same channel have a successful rendezvous is a function of channel states. The channel states are modeled by stochastic processes with joint distributions known to users. However, the exact state of a channel at any time is not observable. We first consider two channel models: (i) the fast time-varying channel model (where the channel states are assumed to be independent and identically distributed in each time slot), and (ii) the slow time-varying channel model (where the channel states remain unchanged over time). Among the classes of the blind rendezvous policies that randomly hop on channels according to certain channel selection probabilities, we show the optimal channel selection policy that minimizes the expected time-to-rendezvous (ETTR) is the single selection policy that hops on the ``best'' channel all the time in the fast time-varying channel model. However, for the slow time-varying channel model, it is much more difficult to find the optimal channel selection policy. By using the majorization ordering, we derive a lower bound and an upper bound for the ETTR under the assumption that the channel states are exchangeable random variables. Bases on these bounds, we then prove various approximation solutions. We then extend our results to general channel models where the joint distribution of the channel states is only assumed to be stationary in time., Comment: 14 pages, 3 figures, under review in IEEE Transactions on Mobile Computing

The network embedding problem aims to map nodes that are similar to each other to vectors in a Euclidean space that are close to each other. Like centrality analysis (ranking) and community detection, network embedding is in general considered as an ill-posed problem, and its solution may depend on a person's view on this problem. In this book chapter, we adopt the framework of sampled graphs that treat a person's view as a sampling method for a network. The modularity for a sampled graph, called the generalized modularity in the book chapter, is a similarity matrix that has a specific probabilistic interpretation. One of the main contributions of this book chapter is to propose using the generalized modularity matrix for network embedding and show that the network embedding problem can be treated as a trace maximization problem like the community detection problem. Our generalized modularity embedding approach is very general and flexible. In particular, we show that the Laplacian eigenmaps is a special case of our generalized modularity embedding approach. Also, we show that dimensionality reduction can be done by using a particular sampled graph. Various experiments are conducted on real datasets to illustrate the effectiveness of our approach., Comment: 11 pages, 7 figures, 3 tables

In this paper, we propose a new averaging model for modeling the competitive influence of $K$ candidates among $n$ voters in an election process. For such an influence propagation model, we address the question of how many seeded voters a candidate needs to place among undecided voters in order to win an election. We show that for a random network generated from the stochastic block model, there exists a percolation threshold for a candidate to win the election if the number of seeded voters placed by the candidate exceeds the threshold. By conducting extensive experiments, we show that our theoretical percolation thresholds are very close to those obtained from simulations for random networks and the errors are within $10\%$ for a real-world network., Comment: 11 pages, 9 figures, this article is the complete version (with proofs) of the IEEE Global Communications Conference 2019 review paper

The multichannel rendezvous problem that asks two secondary users to rendezvous on a common available channel in a cognitive radio network (CRN) has received a lot of attention lately. Most rendezvous algorithms in the literature focused on constructing channel hopping (CH) sequences that guarantee finite maximum time-to-rendezvous (MTTR). However, these algorithms perform rather poorly in terms of the expected time-to-rendezvous (ETTR) even when compared to the simple random algorithm. In this paper, we propose the quasi-random (QR) CH algorithm that has a comparable ETTR to the random algorithm and a comparable MTTR to the best bound in the literature. Our QR algorithm does not require the unique identifier (ID) assumption and it is very simple to implement in the symmetric, asynchronous, and heterogeneous setting with multiple radios. In a CRN with $N$ commonly labelled channels, the MTTR of the QR algorithm is bounded above by $9 M \lceil n_1/m_1 \rceil \cdot \lceil n_2/m_2 \rceil$ time slots, where $n_1$ (resp. $n_2$) is the number of available channels to user $1$ (resp. 2), $m_1$ (resp. $m_2$) is the number of radios for user $1$ (resp. 2), and $M=\lceil \lceil \log_2 N \rceil /4 \rceil *5+6$. Such a bound is only slightly larger than the best $O((\log \log N) \frac{n_1 n_2}{m_1 m_2})$ bound in the literature. When each SU has a single radio, the ETTR is bounded above by $\frac{n_1 n_2}{G}+9Mn_1n_2 \cdot (1-\frac{G}{n_1 n_2})^M$, where $G$ is the number of common channels between these two users. By conducting extensive simulations, we show that for both the MTTR and the ETTR, our algorithm is comparable to the simple random algorithm and it outperforms several existing algorithms in the literature.

In our recent works, we developed a probabilistic framework for structural analysis in undirected networks and directed networks. The key idea of that framework is to sample a network by a symmetric and asymmetric bivariate distribution and then use that bivariate distribution to formerly defining various notions, including centrality, relative centrality, community, and modularity. The main objective of this paper is to extend the probabilistic definition to attributed networks, where sampling bivariate distributions by exponentially twisted sampling. Our main finding is that we find a way to deal with the sampling of the attributed network including signed network. By using the sampling method, we define the various centralities in attributed networks. The influence centralities and trust centralities correctly show that how to identify centralities in signed network. The advertisement-specific influence centralities also perfectly define centralities when the attributed networks that have node attribute. Experimental results on real-world dataset demonstrate the different centralities with changing the temperature. Further experiments are conducted to gain a deeper understanding of the importance of the temperature.

In this paper, we propose a unified framework for sampling, clustering and embedding data points in semi-metric spaces. For a set of data points $\Omega=\{x_1, x_2, \ldots, x_n\}$ in a semi-metric space, we consider a complete graph with $n$ nodes and $n$ self edges and then map each data point in $\Omega$ to a node in the graph with the edge weight between two nodes being the distance between the corresponding two points in $\Omega$. By doing so, several well-known sampling techniques can be applied for clustering data points in a semi-metric space. One particularly interesting sampling technique is the exponentially twisted sampling in which one can specify the desired average distance from the sampling distribution to detect clusters with various resolutions. We also propose a softmax clustering algorithm that can perform a clustering and embed data points in a semi-metric space to a low dimensional Euclidean space. Our experimental results show that after a certain number of iterations of "training", our softmax algorithm can reveal the "topology" of the data from a high dimensional Euclidean. We also show that the eigendecomposition of a covariance matrix is equivalent to the principal component analysis (PCA). To deal with the hierarchical structure of clusters, our softmax clustering algorithm can also be used with a hierarchical clustering algorithm. For this, we propose a partitional-hierarchical algorithm, called $i$PHD, in this paper. Our experimental results show that those algorithms based on the maximization of normalized modularity tend to balance the sizes of detected clusters and thus do not perform well when the ground-truth clusters are different in sizes. Also, using a metric is better than using a semi-metric as the triangular inequality is not satisfied for a semi-metric and that is more prone to clustering errors.

In this paper, we consider the community detection problem in signed networks, where there are two types of edges: positive edges (friends) and negative edges (enemies). One renowned theorem of signed networks, known as Harary's theorem, states that structurally balanced signed networks are clusterable. By viewing each cycle in a signed network as a parity-check constraint, we show that the community detection problem in a signed network with two communities is equivalent to the decoding problem for a parity-check code. We also show how one can use two renowned decoding algorithms in error- correcting codes for community detection in signed networks: the bit-flipping algorithm, and the belief propagation algorithm. In addition to these two algorithms, we also propose a new community detection algorithm, called the Hamming distance algorithm, that performs community detection by finding a codeword that minimizes the Hamming distance. We compare the performance of these three algorithms by conducting various experiments with known ground truth. Our experimental results show that our Hamming distance algorithm outperforms the other two.

In this paper, we first propose a new iterative algorithm, called the K-sets+ algorithm for clustering data points in a semi-metric space, where the distance measure does not necessarily satisfy the triangular inequality. We show that the K-sets+ algorithm converges in a finite number of iterations and it retains the same performance guarantee as the K-sets algorithm for clustering data points in a metric space. We then extend the applicability of the K-sets+ algorithm from data points in a semi-metric space to data points that only have a symmetric similarity measure. Such an extension leads to great reduction of computational complexity. In particular, for an n * n similarity matrix with m nonzero elements in the matrix, the computational complexity of the K-sets+ algorithm is O((Kn + m)I), where I is the number of iterations. The memory complexity to achieve that computational complexity is O(Kn + m). As such, both the computational complexity and the memory complexity are linear in n when the n * n similarity matrix is sparse, i.e., m = O(n). We also conduct various experiments to show the effectiveness of the K-sets+ algorithm by using a synthetic dataset from the stochastic block model and a real network from the WonderNetwork website.

The matrix factorization (MF) technique has been widely adopted for solving the rating prediction problem in recommender systems. The MF technique utilizes the latent factor model to obtain static user preferences (user latent vectors) and item characteristics (item latent vectors) based on historical rating data. However, in the real world user preferences are not static but full of dynamics. Though there are several previous works that addressed this time varying issue of user preferences, it seems (to the best of our knowledge) that none of them is specifically designed for tracking concept drift in individual user preferences. Motivated by this, we develop a Temporal Matrix Factorization approach (TMF) for tracking concept drift in each individual user latent vector. There are two key innovative steps in our approach: (i) we develop a modified stochastic gradient descent method to learn an individual user latent vector at each time step, and (ii) by the Lasso regression we learn a linear model for the transition of the individual user latent vectors. We test our method on a synthetic dataset and several real datasets. In comparison with the original MF, our experimental results show that our temporal method is able to achieve lower root mean square errors (RMSE) for both the synthetic and real datasets. One interesting finding is that the performance gain in RMSE is mostly from those users who indeed have concept drift in their user latent vectors at the time of prediction. In particular, for the synthetic dataset and the Ciao dataset, there are quite a few users with that property and the performance gains for these two datasets are roughly 20% and 5%, respectively.

In our recent works, we developed a probabilistic framework for structural analysis in undirected networks. The key idea of that framework is to sample a network by a symmetric bivariate distribution and then use that bivariate distribution to formerly define various notions, including centrality, relative centrality, community, and modularity. The main objective of this paper is to extend the probabilistic framework to directed networks, where the sampling bivariate distributions could be asymmetric. Our main finding is that we can relax the assumption from symmetric bivariate distributions to bivariate distributions that have the same marginal distributions. By using such a weaker assumption, we show that various notions for structural analysis in directed networks can also be defined in the same manner as before. However, since the bivariate distribution could be asymmetric, the community detection algorithms proposed in our previous work cannot be directly applied. For this, we show that one can construct another sampled graph with a symmetric bivariate distribution so that for any partition of the network, the modularity index remains the same as that of the original sampled graph. Based on this, we propose a hierarchical agglomerative algorithm that returns a partition of communities when the algorithm converges.

Clustering is one of the most fundamental problems in data analysis and it has been studied extensively in the literature. Though many clustering algorithms have been proposed, clustering theories that justify the use of these clustering algorithms are still unsatisfactory. In particular, one of the fundamental challenges is to address the following question: What is a cluster in a set of data points? In this paper, we make an attempt to address such a question by considering a set of data points associated with a distance measure (metric). We first propose a new cohesion measure in terms of the distance measure. Using the cohesion measure, we define a cluster as a set of points that are cohesive to themselves. For such a definition, we show there are various equivalent statements that have intuitive explanations. We then consider the second question: How do we find clusters and good partitions of clusters under such a definition? For such a question, we propose a hierarchical agglomerative algorithm and a partitional algorithm. Unlike standard hierarchical agglomerative algorithms, our hierarchical agglomerative algorithm has a specific stopping criterion and it stops with a partition of clusters. Our partitional algorithm, called the K-sets algorithm in the paper, appears to be a new iterative algorithm. Unlike the Lloyd iteration that needs two-step minimization, our K-sets algorithm only takes one-step minimization. One of the most interesting findings of our paper is the duality result between a distance measure and a cohesion measure. Such a duality result leads to a dual K-sets algorithm for clustering a set of data points with a cohesion measure. The dual K-sets algorithm converges in the same way as a sequential version of the classical kernel K-means algorithm. The key difference is that a cohesion measure does not need to be positive semi-definite.

In this two-part paper, we consider SDL constructions of optical queues with a limited number of recirculations through the optical switches and the fiber delay lines. We show that the constructions of certain types of optical queues, including linear compressors, linear decompressors, and 2-to-1 FIFO multiplexers, under a simple packet routing scheme and under the constraint of a limited number of recirculations can be transformed into equivalent integer representation problems under a corresponding constraint. Given $M$ and $k$, the problem of finding an \emph{optimal} construction, in the sense of maximizing the maximum delay (resp., buffer size), among our constructions of linear compressors/decompressors (resp., 2-to-1 FIFO multiplexers) is equivalent to the problem of finding an optimal sequence ${\dbf^*}_1^M$ in $\Acal_M$ (resp., $\Bcal_M$) such that $B({\dbf^*}_1^M;k)=\max_{\dbf_1^M\in \Acal_M}B(\dbf_1^M;k)$ (resp., $B({\dbf^*}_1^M;k)=\max_{\dbf_1^M\in \Bcal_M}B(\dbf_1^M;k)$), where $\Acal_M$ (resp., $\Bcal_M$) is the set of all sequences of fiber delays allowed in our constructions of linear compressors/decompressors (resp., 2-to-1 FIFO multiplexers). In Part I, we propose a class of \emph{greedy} constructions of linear compressors/decompressors and 2-to-1 FIFO multiplexers by specifying a class $\Gcal_{M,k}$ of sequences such that $\Gcal_{M,k}\subseteq \Bcal_M\subseteq \Acal_M$ and each sequence in $\Gcal_{M,k}$ is obtained recursively in a greedy manner. We then show that every optimal construction must be a greedy construction. In Part II, we further show that there are at most two optimal constructions and give a simple algorithm to obtain the optimal construction(s)., Comment: 59 pages; 1 figure; This paper was presented in part at the IEEE International Conference on Computer Communications (INFOCOM'08), Phoenix, AZ, USA, April~13--18, 2008. This paper has been submitted to IEEE Transactions on Information Theory for possible publication.