Angsheng Li

The small-community phenomenon in networks

We investigate several geometric models of networks that simultaneously have some nice global properties, including the small-diameter property, the small-community phenomenon, which is defined to capture the common experience that (almost) everyone in society also belongs to some meaningful small communities, and the power law degree distribution, for which our result significantly strengthens those given in van den Esker (2008) and Jordan (2010). These results, together with our previous work in Li and Peng (2011), build a mathematical foundation for the study of both communities and the small-community phenomenon in various networks. In the proof of the power law degree distribution, we develop the method of alternating concentration analysis to build a concentration inequality by alternately and iteratively applying both the sub-and super-martingale inequalities, which seems to be a powerful technique with further potential applications.

Community Structures in Classical Network Models

Abstract Communities (or clusters) are ubiquitous in real-world networks. Researchers from different fields have proposed many definitions of communities, which are usually thought of as a subset of nodes whose vertices are well connected with other vertices in the set and have relatively fewer connections with vertices outside the set. In contrast to traditional research that focuses mainly on detecting and/or testing such clusters, we propose a new definition of community and a novel way to study community structure, with which we are able to investigate mathematical network models to test whether they exhibit the small-community phenomenon, i.e., whether every vertex in the network belongs to some small community. We examine various models and establish both positive and negative results: we show that in some models, the small-community phenomenon exists, while in some other models, it does not.

Turing Definability in the Ershov Hierarchy

Univ Leeds, Sch Math, Dept Pure Math, Leeds LS2 9JT, W Yorkshire, England. Chinese Acad Sci, Inst Software, Beijing 100080, Peoples R China.

Algorithmic aspects of homophyly of networks

Abstract We investigate the algorithmic problems of the homophyly phenomenon in networks. Given an undirected graph G = ( V , E ) and a vertex coloring c : V → { 1 , 2 , ⋯ , k } of G , we say that a vertex v ∈ V is happy if v shares the same color with all its neighbors, and unhappy , otherwise, and that an edge e ∈ E is happy , if its two endpoints have the same color, and unhappy , otherwise. Supposing c is a partial vertex coloring of G , we define the Maximum Happy Vertices problem (MHV, for short) as to color all the remaining vertices such that the number of happy vertices is maximized, and the Maximum Happy Edges problem (MHE, for short) as to color all the remaining vertices such that the number of happy edges is maximized. Let k be the number of colors allowed in the problems. We show that both MHV and MHE can be solved in polynomial time if k = 2 , and that both MHV and MHE are NP-hard if k ≥ 3 . We devise a max ⁡ { 1 / k , Ω ( Δ − 3 ) } -approximation algorithm for the MHV problem, where Δ is the maximum degree of vertices in the input graph, and a 1/2-approximation algorithm for the MHE problem. This is the first theoretical progress of these two natural and fundamental new problems.

Structural Information and Dynamical Complexity of Networks

In 1953, Shannon proposed the question of quantification of structural information to analyze communication systems. The question has become one of the longest great challenges in information science and computer science. Here, we propose the first metric for structural information. Given a graph G , we define the K-dimensional structural information of G (or structure entropy of G), denoted by HK(G) , to be the minimum overall number of bits required to determine the K-dimensional code of the node that is accessible from random walk in G. The K-dimensional structural information provides the principle for completely detecting the natural or true structure, which consists of the rules, regulations, and orders of the graphs, for fully distinguishing the order from disorder in structured noisy data, and for analyzing communication systems, solving the Shannons problem and opening up new directions. The K-dimensional structural information is also the first metric of dynamical complexity of networks, measuring the complexity of interactions, communications, operations, and even evolution of networks. The metric satisfies a number of fundamental properties, including additivity, locality, robustness, local and incremental computability, and so on. We establish the fundamental theorems of the one- and two-dimensional structural information of networks, including both lower and upper bounds of the metrics of classic data structures, general graphs, the networks of models, and the networks of natural evolution. We propose algorithms to approximate the K-dimensional structural information of graphs by finding the K-dimensional structure of the graphs that minimizes the K-dimensional structure entropy. We find that the K-dimensional structure entropy minimization is the principle for detecting the natural or true structures in real-world networks. Consequently, our structural information provides the foundation for knowledge discovering from noisy data. We establish a black hole principle by using the two-dimensional structure information of graphs. We propose the natural rank of locally listing algorithms by the structure entropy minimization principle, providing the basis for a next-generation search engine.

Homophyly/Kinship Model: Naturally Evolving Networks

It has been a challenge to understand the formation and roles of social groups or natural communities in the evolution of species, societies and real world networks. Here, we propose the hypothesis that homophyly/kinship is the intrinsic mechanism of natural communities, introduce the notion of the affinity exponent and propose the homophyly/kinship model of networks. We demonstrate that the networks of our model satisfy a number of topological, probabilistic and combinatorial properties and, in particular, that the robustness and stability of natural communities increase as the affinity exponent increases and that the reciprocity of the networks in our model decreases as the affinity exponent increases. We show that both homophyly/kinship and reciprocity are essential to the emergence of cooperation in evolutionary games and that the homophyly/kinship and reciprocity determined by the appropriate affinity exponent guarantee the emergence of cooperation in evolutionary games, verifying Darwin’s proposal that kinship and reciprocity are the means of individual fitness. We propose the new principle of structure entropy minimisation for detecting natural communities of networks and verify the functional module property and characteristic properties by a healthy tissue cell network, a citation network, some metabolic networks and a protein interaction network.

Strategies for network security

Security of networks has become an increasingly important issue in the highly connected world. Security depends on attacks. Typical attacks include both cascading failure of virus spreading and of information propagation and physical attacks of removal of nodes or edges. Numerous experiments have shown that none of the existing models construct secure networks, and that the universal properties of power law and small world phenomenon of networks seem unavoidable obstacles for security of networks against attacks. Here, we propose a new strategy of attacks, the attack of rules of evolution of networks. By using the strategy, we proposed a new model of networks which generates provably secure networks. It was shown both analytically and numerically that the best strategy is to attack on the rules of the evolution of networks, that the second best strategy is the attack by cascading failure models, that the third best strategy is the physical attack of removal of nodes or edges, and that the least desirable strategy is the physical attack of deleting structures of the networks. The results characterize and classify the strategies for network security, providing a foundation for a security theory of networks. Equally important, our results demonstrate that security can be achieved provably by structures of networks, that there is a tradeoff between the roles of structures and of thresholds in security engineering, and that power law and small world property are never obstacles of security of networks. Our model explores a number of new principles of networks, including some topological principles, probabilistic principles, and combinatorial principles. The new principles build the foundation for new strategies for enhancing security of networks, and for new protocols of communications and security of the Internet and computer networks etc. We anticipate that our theory can be used in analyzing security of real systems in economy, society and technology.

Preface to special issue: Theory and applications of models of computation (tamc)

Theory and Applications of Models of Computation (TAMC) is an international conference series with an interdisciplinary character bringing together researchers working in computer science, mathematics (especially logic) and the physical sciences. This interdisciplinary approach, with an emphasis on the theory of computation in a broad sense, gives the series its special appeal within China and internationally. At a time when the pressures are increasingly towards narrowly ad hoc research, and scientific fragmentation, meetings that reassert the importance of theory, fundamental concepts and a wider perspective have an important role to play.

Improved Approximation Algorithms for the Maximum Happy Vertices and Edges Problems

The Maximum Happy Vertices (MHV) problem and the Maximum Happy Edges (MHE) problem are two fundamental problems arising in the study of the homophyly phenomenon in large scale networks. Both of these two problems are NP-hard. Interestingly, the MHE problem is a natural generalization of Multiway Uncut, the complement of the classic Multiway Cut problem. In this paper, we present new approximation algorithms for MHV and MHE based on randomized LP-rounding techniques. Specifically, we show that MHV can be approximated within \(\frac{1}{\varDelta +1}\), where \({\varDelta }\) is the maximum vertex degree, and MHE can be approximated within \(\frac{1}{2} + \frac{\sqrt{2}}{4}f(k) \ge 0.8535\), where \(f(k) \ge 1\) is a function of the color number k. These results improve on the previous approximation ratios for MHV, MHE as well as Multiway Uncut in the literature.

The complexity and approximability of minimum contamination problems

In this article, we investigate the complexity and approximability of the Minimum Contamination Problems, which are derived from epidemic spreading areas and have been extensively studied recently. We show that both the Minimum Average Contamination Problem and the Minimum Worst Contamination Problem are NP-hard problems even on restrict cases. For any e > 0, we give (1 + e,O(1+e/elog n))-bicriteria approximation algorithm for the Minimum Average Contamination Problem. Moreover, we show that theMinimumAverage Contamination Problem is NP-hard to be approximated within 5/3- e and the Minimum Worst Contamination Problem is NP-hard to be approximated within 2 - e, for any e > 0, giving the first hardness results of approximation of constant ratios to the problems.