抗原变异可使得多种致病微生物易于逃避宿主免疫应答。表达在感染红细胞表面的恶性疟原虫红细胞表面蛋白1（PfPMP1）与感染红细胞、内皮细胞、树突状细胞以及胎盘的单个或多个受体作用，在黏附及免疫逃避中起关键的作用。每个单倍体基因组var基因家族编码约60种成员，通过启动转录不同的var基因变异体为抗原变异提供了分子基础。

疟原虫var基因转换速率变化导致抗原变异[英]／Paul H, Robert P, Christodoulou Z, et al//Proc Natl Acad Sci U S A

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Introduction. 1. Mathematical Preliminaries. Submodular Systems and Base Polyhedra. 2. From Matroids to Submodular Systems. Matroids. Polymatroids. Submodular Systems. 3. Submodular Systems and Base Polyhedra. Fundamental Operations on Submodular Systems. Greedy Algorithm. Structures of Base Polyhedra. Intersecting- and Crossing-Submodular Functions. Related Polyhedra. Submodular Systems of Network Type. Neoflows. 4. The Intersection Problem. The Intersection Theorem. The Discrete Separation Theorem. The Common Base Problem. 5. Neoflows. The Equivalence of the Neoflow Problems. Feasibility for Submodular Flows. Optimality for Submodular Flows. Algorithms for Neoflows. Matroid Optimization. Submodular Analysis. 6. Submodular Functions and Convexity. Conjugate Functions and a Fenchel-Type Min-Max Theorem for Submodular and Supermodular Functions. Subgradients of Submodular Functions. The Lovasz Extensions of Submodular Functions. 7. Submodular Programs. Submodular Programs - Unconstrained Optimization. Submodular Programs - Constrained Optimization. Nonlinear Optimization with Submodular Constraints. 8. Separable Convex Optimization. Optimality Conditions. A Decomposition Algorithm. Discrete Optimization. 9. The Lexicographically Optimal Base Problem. Nonlinear Weight Functions. Linear Weight Functions. 10. The Weighted Max-Min and Min-Max Problems. Continuous Variables. Discrete Variables. 11. The Fair Resource Allocation Problem. Continuous Variables. Discrete Variables. 12. The Neoflow Problem with a Separable Convex Cost Function. References. Index.

Submodular functions and optimization

Real-world complex networks are composed of non-random quantitative interactions. Identifying communities of nodes that tend to interact more with each other than the network as a whole is a key research focus across multiple disciplines, yet many community detection algorithms only use information about the presence or absence of interactions between nodes. Weighted modularity is a potential method for evaluating the quality of community partitions in quantitative networks. In this framework, the optimal community partition of a network can be found by searching for the partition that maximizes modularity. Attempting to find the partition that maximizes modularity is a computationally hard problem requiring the use of algorithms. QuanBiMo is an algorithm that has been proposed to maximize weighted modularity in bipartite networks. This paper introduces two new algorithms, LPAwb+ and DIRTLPAwb+, for maximizing weighted modularity in bipartite networks. LPAwb+ and DIRTLPAwb+ robustly identify partitions with high modularity scores. DIRTLPAwb+ consistently matched or outperformed QuanBiMo, while the speed of LPAwb+ makes it an attractive choice for detecting the modularity of larger networks. Searching for modules using weighted data (rather than binary data) provides a different and potentially insightful method for evaluating network partitions.

https://royalsocietypublishing.org/doi/pdf/10.1098/rsos.140536

Improved community detection in weighted bipartite networks

Detecting and analyzing dense groups or communities from social and information networks has attracted immense attention over the last decade due to its enormous applicability in different domains. Community detection is an ill-defined problem, as the nature of the communities is not known in advance. The problem has turned even more complicated due to the fact that communities emerge in the network in various forms such as disjoint, overlapping, and hierarchical. Various heuristics have been proposed to address these challenges, depending on the application in hand. All these heuristics have been materialized in the form of new metrics, which in most cases are used as optimization functions for detecting the community structure, or provide an indication of the goodness of detected communities during evaluation. Over the last decade, a large number of such metrics have been proposed. Thus, there arises a need for an organized and detailed survey of the metrics proposed for community detection and evaluation. Here, we present a survey of the start-of-the-art metrics used for the detection and the evaluation of community structure. We also conduct experiments on synthetic and real networks to present a comparative analysis of these metrics in measuring the goodness of the underlying community structure.

Metrics for Community Analysis: A Survey

Identifying community structure in networks is an issue of particular interest in network science. The modularity introduced by Newman and Girvan is the most popular quality function for community detection in networks. In this study, we identify a problem in the concept of modularity and suggest a solution to overcome this problem. Specifically, we obtain a new quality function for community detection. We refer to the function as Z-modularity because it measures the Z-score of a given partition with respect to the fraction of the number of edges within communities. Our theoretical analysis shows that Z-modularity mitigates the resolution limit of the original modularity in certain cases. Computational experiments using both artificial networks and well-known real-world networks demonstrate the validity and reliability of the proposed quality function.

/pdf/z-score-based-modularity-for-community-detection-in-networks-4x1rlonmpy.pdf

Z-score-based modularity for community detection in networks

A hypergraph is a useful combinatorial object to model ternary or higher-order relations among entities. Clustering hypergraphs is a fundamental task in network analysis. In this study, we develop two clustering algorithms based on personalized PageRank on hypergraphs. The first one is local in the sense that its goal is to find a tightly connected vertex set with a bounded volume including a specified vertex. The second one is global in the sense that its goal is to find a tightly connected vertex set. For both algorithms, we discuss theoretical guarantees on the conductance of the output vertex set. Also, we experimentally demonstrate that our clustering algorithms outperform existing methods in terms of both the solution quality and running time. To the best of our knowledge, ours are the first practical algorithms for hypergraphs with theoretical guarantees on the conductance of the output set.

Hypergraph Clustering Based on PageRank

In the densest subgraph problem, given an edge-weighted undirected graph $$G=(V,E,w)$$
 , we are asked to find $$S\subseteq V$$
 that maximizes the density, i.e., w(S) / |S|, where w(S) is the sum of weights of the edges in the subgraph induced by S. This problem has often been employed in a wide variety of graph mining applications. However, the problem has a drawback; it may happen that the obtained subset is too large or too small in comparison with the size desired in the application at hand. In this study, we address the size issue of the densest subgraph problem by generalizing the density of $$S\subseteq V$$
 . Specifically, we introduce the f-density of $$S\subseteq V$$
 , which is defined as w(S) / f(|S|), where $$f:\mathbb {Z}_{\ge 0}\rightarrow \mathbb {R}_{\ge 0}$$
 is a monotonically non-decreasing function. In the f-densest subgraph problem (f-DS), we aim to find $$S\subseteq V$$
 that maximizes the f-density w(S) / f(|S|). Although f-DS does not explicitly specify the size of the output subset of vertices, we can handle the above size issue using a convex/concave size function f appropriately. For f-DS with convex function f, we propose a nearly-linear-time algorithm with a provable approximation guarantee. On the other hand, for f-DS with concave function f, we propose an LP-based exact algorithm, a flow-based $$O(|V|^3)$$
 -time exact algorithm for unweighted graphs, and a nearly-linear-time approximation algorithm.

/pdf/the-densest-subgraph-problem-with-a-convex-concave-size-u26ds6c440.pdf

The Densest Subgraph Problem with a Convex/Concave Size Function

The clique partitioning problem is among the most fundamental graph partitioning problems. Given a complete weighted undirected graph, we find a vertex-disjoint union of cliques so as to maximize the sum of the weights within the cliques. The problem is known to be NP-hard, and exact algorithms and heuristics have been developed so far. In 1989, Grotschel and Wakabayashi (Math Program 45:59–96, 1989) proposed an integer linear programming formulation for the problem. Although this formulation has been employed by many algorithms, it is prohibitive even for relatively small graphs because it contains numerous constraints called transitivity constraints. In this study, we theoretically derive a certain class of redundant constraints in the formulation. Furthermore, we show that the constraints in the derived class are also redundant in the linear programming relaxation of the formulation. By using our results, the number of constraints treated in exact algorithms and heuristics may be reduced. We confirmed that more than half of the transitivity constraints are redundant for some instances, and that computation times to find an optimal solution are shortened for most instances.

Redundant constraints in the standard formulation for the clique partitioning problem

Cheeger's inequality states that a tightly connected subset can be extracted from a graph $G$ using an eigenvector of the normalized Laplacian associated with $G$ More specifically, we can compute a subset with conductance $O(\sqrt{\phi_G})$, where $\phi_G$ is the minimum conductance of a set in $G$ It has recently been shown that Cheeger's inequality can be extended to hypergraphs However, as the normalized Laplacian of a hypergraph is no longer a matrix, we can only approximate to its eigenvectors; this causes a loss in the conductance of the obtained subset To address this problem, we here consider the heat equation on hypergraphs, which is a differential equation exploiting the normalized Laplacian We show that the heat equation has a unique solution and that we can extract a subset with conductance $\sqrt{\phi_G}$ from the solution An analogous result also holds for directed graphs

Atsushi Miyauchi

Papers

Z-score-based modularity for community detection in networks

Hypergraph Clustering Based on PageRank

The Densest Subgraph Problem with a Convex/Concave Size Function

Redundant constraints in the standard formulation for the clique partitioning problem

Finding Cheeger Cuts in Hypergraphs via Heat Equation.