Yicheng Pan

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.

Three-Dimensional Gene Map of Cancer Cell Types: Structural Entropy Minimisation Principle for Defining Tumour Subtypes

In this study, we propose a method for constructing cell sample networks from gene expression profiles, and a structural entropy minimisation principle for detecting natural structure of networks and for identifying cancer cell subtypes. Our method establishes a three-dimensional gene map of cancer cell types and subtypes. The identified subtypes are defined by a unique gene expression pattern, and a three-dimensional gene map is established by defining the unique gene expression pattern for each identified subtype for cancers, including acute leukaemia, lymphoma, multi-tissue, lung cancer and healthy tissue. Our three-dimensional gene map demonstrates that a true tumour type may be divided into subtypes, each defined by a unique gene expression pattern. Clinical data analyses demonstrate that most cell samples of an identified subtype share similar survival times, survival indicators and International Prognostic Index (IPI) scores and indicate that distinct subtypes identified by our algorithms exhibit different overall survival times, survival ratios and IPI scores. Our three-dimensional gene map establishes a high-definition, one-to-one map between the biologically and medically meaningful tumour subtypes and the gene expression patterns, and identifies remarkable cells that form singleton submodules.

A Theory of Network Security: Principles of Natural Selection and Combinatorics

We propose the definition of security of networks against the cascading failure models of deliberate attacks. We propose a model of networks by the natural selection of homophyly/kinship, randomness and preferential attachment, referred to as security model. We show that the networks generated by the security model are provably secure against any attacks of sizes poly(log n) under the cascading failure models, for which the principles of natural selection and the combinatorial principles of the networks of the security model, including a power law, a self-organizing principle, a small diameter property, a local navigation law, a degree priority principle, an inclusion-exclusion principle, and an infection priority tree principle etc, are the underlying principles. Furthermore, we show that the networks generated by the security model have an expander core. This property ensures that the networks of the security model satisfy the requirement of global communications in engineering. Based on our theory, we propose a security protocol for computer networks. Our theory demonstrates that security of networks can be achieved by a merging of natural selection and combinatorial principles, and that both natural selection principle and combinatorial principles are essential to security of networks.

Resistance and Security Index of Networks: Structural Information Perspective of Network Security

Recently, Li and Pan defined the metric of the K-dimensional structure entropy of a structured noisy dataset G to be the information that controls the formation of the K-dimensional structure of G that is evolved by the rules, order and laws of G, excluding the random variations that occur in G. Here, we propose the notion of resistance of networks based on the one- and two-dimensional structural information of graphs. Given a graph G, we define the resistance of G, written , as the greatest overall number of bits required to determine the code of the module that is accessible via random walks with stationary distribution in G, from which the random walks cannot escape. We show that the resistance of networks follows the resistance law of networks, that is, for a network G, the resistance of G is , where and are the one- and two-dimensional structure entropies of G, respectively. Based on the resistance law, we define the security index of a network G to be the normalised resistance of G, that is, . We show that the resistance and security index are both well-defined measures for the security of the networks.

Global core, and galaxy structure of networks

We propose a novel approach, namely local reduction of networks, to extract the global core (GC, for short) from a complex network. The algorithm is built based on the small community phenomenon of networks. The global cores found by our local reduction from some classical graphs and benchmarks convince us that the global core of a network is intuitively the supporting graph of the network, which is “similar to” the original graph, that the global core is small and essential to the global properties of the network, and that the global core, together with the small communities gives rise to a clear picture of the structure of the network, that is, the galaxy structure of networks. We implement the local reduction to extract the global cores for a series of real networks, and execute a number of experiments to analyze the roles of the global cores for various real networks. For each of the real networks, our experiments show that the found global core is small, that the global core is similar to the original network in the sense that it follows the power law degree distribution with power exponent close to that of the original network, that the global core is sensitive to errors for both cascading failure and physical attack models, in the sense that a small number of random errors in the global core may cause a major failure of the whole network, and that the global core is a good approximate solution to the r-radius center problem, leading to a galaxy structure of the network.

Establishing social cooperation: The role of hubs and community structure

Prisoner’s Dilemma games have become a well-established paradigm for studying the mechanisms by which cooperative behaviour may evolve in societies consisting of selfish individuals. Recent research has focussed on the effect of spatial and connectivity structure in promoting the emergence of cooperation in scenarios where individuals play games with their neighbors, using simple ‘memoryless’ rules to decide their choice of strategy in repeated games. While heterogeneity and structural features such as clustering have been seen to lead to reasonable levels of cooperation in very restricted settings, no conditions on network structure have been established which robustly ensure the emergence of cooperation in a manner which is not overly sensitive to parameters such as network size, average degree, or the initial proportion of cooperating individuals. Here we consider a natural random network model, with parameters which allow us to vary the level of ‘community’ structure in the network, as well as the number of high degree hub nodes. We investigate the effect of varying these structural features and show that, for appropriate choices of these parameters, cooperative behaviour does now emerge in a truly robust fashion and to a previously unprecedented degree. The implication is that cooperation (as modelled here by Prisoner’s Dilemma games) can become the social norm in societal structures divided into smaller communities, and in which hub nodes provide the majority of inter-community connections.